FRM Quant Question Bank

Topic 15 - Probabilities

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A company says that whether its earnings increase depends on whether it increased its dividends. From this we know:

P(dividend increase or earnings increase) = P(both dividend and earnings increase).

P(both dividend increase and earnings increase) = P(dividend increase). P(dividend increase | earnings increase) is not equal to P(earnings increase). P(earnings increase | dividend increase) is not equal to P(earnings increase). Explanation

If two events A and B are dependent, then the conditional probabilities of P(A|B) and P(B|A) will not equal their respective unconditional probabilities (of P(A) and P(B), respectively). The other choices may or may not occur, e.g., P(A | B) = P(B) is possible but not necessary.

A conditional expectation involves:

determining the expected joint probability.

estimating the skewness.

calculating the conditional variance.

refining a forecast because of the occurrence of some other event. Explanation

Conditional expected values are contingent upon the occurrence of some other event. The expectation changes as new information is revealed.

An investor is choosing one of twenty securities. Ten of the securities are stocks and ten are bonds. Four of the ten stocks were issued by utilities, the other six were issued by industrial firms. Two of the ten bonds were issued by utilities, the other eight were issued by industrial firms. If the investor chooses a security at random, the probability that it is a bond or a security issued by an industrial firm is:

0.80.

0.50. 0.60. 0.70. Explanation

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bond=P(B)=10/20, P(I) is the probability that a security was issued by an industrial firm=P(I)=14/20, and P(B∩I) is the probability that a security is both a bond and issued by an industrial firm=P(B∩I)=8/20. 10/20+14/20-8/20=16/20=0.80.

A joint probability of A and B must always be:

less than or equal to the conditional probability of A given B.

greater than or equal to than the probability of A or B.

greater than or equal to the conditional probability of A given B. less than the probability of A and the probability of B.

Explanation

By the formula for joint probability: P(AB)=P(A|B) × P(B), since P(B) ≤ 1, then P(AB) ≤P(A|B). None of the other choices must hold.

Dependent random variables are defined as variables where their joint probability is:

not equal to the product of their individual probabilities.

equal to zero.

equal to the product of their individual probabilities. greater than the product of their individual probabilities. Explanation

Dependence results between random variables when their joint probabilities are not equal products of individual probabilities. If they are equal, then an independent relationship exists.

Thomas Baynes has applied to both Harvard and Yale. Baynes has determined that the probability of getting into Harvard is 25% and the probability of getting into Yale (his father's alma mater) is 42%. Baynes has also determined that the probability of being accepted at both schools is 2.8%. What is the probability of Baynes being accepted at either Harvard or Yale?

7.7%.

67.0%. 10.5%. 64.2%. Explanation

Using the addition rule, the probability of being accepted at Harvard or Yale, is equal to: P(Harvard) + P(Yale) − P(Harvard and Yale) = 0.25 + 0.42 − 0.028 = 0.642 or 64.2%.

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The probability of a new Wal-Mart being built in town is 64%. If Wal-Mart comes to town, the probability of a new Wendy's restaurant being built is 90%. What is the probability of a new Wal-Mart and a new Wendy's restaurant being built?

0.306. 0.675. 0.240. 0.576. Explanation P(AB) = P(A|B) × P(B)

The probability of a new Wal-Mart and a new Wendy's is equal to the probability of a new Wendy's "if Wal-Mart" (0.90) times the probability of a new Wal-Mart (0.64). (0.90)(0.64) = 0.576.

A dealer in a casino has rolled a five on a single die three times in a row. What is the probability of her rolling another five on the next roll, assuming it is a fair die?

0.200.

0.500. 0.167. 0.001. Explanation

The probability of a value being rolled is 1/6 regardless of the previous value rolled.

X and Y are discrete random variables. The probability that X = 3 is 0.20 and the probability that Y = 4 is 0.30. The probability of observing that X = 3 and Y = 4 concurrently is closest to:

0.06.

0. 0.50.

Cannot answer with the information provided. Explanation

If you knew that X and Y were independent, you could calculate the probability as 0.20(0.30) = 0.06. Without this knowledge, you would need the joint probability distribution.

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Which of the following statements about probability is most accurate?

An outcome is the calculated probability of an event.

Out of a sample of 100 widgets 10 were found to be defective, 20 were perfect, and 70 were OK. The probability of picking a perfect widget at random is 29%.

An event is a set of one or more possible values of a random variable.

A conditional probability is the probability that two or more events will happen concurrently. Explanation

Conditional probability is the probability of one event happening given that another event has happened. An outcome is the numerical result associated with a random variable. The probability of picking a perfect widget is 20 / 100 = 0.20 or 20%.

The probabilities that three students will earn an A on an exam are 0.20, 0.25, and 0.30, respectively. If each student's performance is independent of that of the other two students, the probability that all three students will earn an A is closest to:

0.0150.

0.0075. 0.0010. 0.7500. Explanation

Since the events are independent, the probability is obtained by multiplying the individual probabilities. Probability of three As = (0.20) × (0.25) × (0.30) = 0.0150.

An investor has an A-rated bond, a BB-rated bond, and a CCC-rated bond where the probabilities of default over the next three years are 4 percent, 12 percent, and 30 percent, respectively. What is the probability that all of these bonds will default in the next three years if the individual default probabilities are independent?

1.44%.

0.14%. 46.00%. 23.00%. Explanation

Since the probability of default for each bond is independent, P(A BB CCC) = P(A) × P(BB) × P(CCC) = 0.04 × 0.12 × 0.30 = 0.00144 = 0.14%.

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mutually exclusive. statistically independent. conditionally dependent. collectively exhaustive.

Explanation

If the outcome of one event does not influence the outcome of another, then the events are independent.

If the probability of both a new Wal-Mart and a new Wendy's being built next month is 68% and the probability of a new Wal-Mart being built is 85%, what is the probability of a new Wendy's being built if a new Wal-Mart is built?

0.85. 0.80. 0.70. 0.60. Explanation P(AB) = P(A|B) × P(B) 0.68 / 0.85 = 0.80

If X and Y are independent events, which of the following is most accurate?

P(X or Y) = (P(X)) × (P(Y)).

P(X | Y) = P(X).

X and Y cannot occur together. P(X or Y) = P(X) + P(Y). Explanation

Note that events being independent means that they have no influence on each other. It does not necessarily mean that they are mutually exclusive. Accordingly, P(X or Y) = P(X) + P(Y) − P(X and Y). By the definition of independent events, P(X|Y) = P(X).

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✓ A) ✗ B) ✗ C) ✗ D) P(A and B) = 0.08. P(A or B) = 0.52. P(A and B) = 0. P(B|A) = 0.20. Explanation

If the two evens are mutually exclusive, the probability of both ocurring is zero.

The following table summarizes the availability of trucks with air bags and bucket seats at a dealership.

Bucket

seats

No Bucket

Seats

Total

What is the probability of randomly selecting a truck with air bags and bucket seats?

0.34. 0.57. 0.28. 0.16. Explanation 75 / 220 = 0.34.

The following table summarizes the availability of trucks with air bags and bucket seats at a dealership. Bucket Seats No Bucket Seats Total

Air Bags 75 50 125

No Air Bags 35 60 95

Total 110 110 220

What is the probability of selecting a truck at random that has either air bags or bucket seats?

73%.

50%. 34%. 107%. Explanation

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The addition rule for probabilities is used to determine the probability of at least one event among two or more events occurring. The probability of each event is added and the joint probability (if the events are not mutually exclusive) is subtracted to arrive at the solution. P(air bags or bucket seats) = P(air bags) + P(bucket seats) − P(air bags and bucket seats) = (125 / 220) + (110 / 220) − (75 / 220) = 0.57 + 0.50 − 0.34 = 0.73 or 73%.

Alternative: 1 − P(no airbag and no bucket seats) = 1 − (60 / 220) = 72.7%

If two fair coins are flipped and two fair six-sided dice are rolled, all at the same time, what is the probability of ending up with two heads (on the coins) and two sixes (on the dice)?

0.8333.

0.4167. 0.0039. 0.0069. Explanation

For the four independent events defined here, the probability of the specified outcome is 0.5000 × 0.5000 × 0.1667 × 0.1667 = 0.0069.

Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of both suffering from allergies and not suffering from allergies?

Suffer from Allergies Don't Suffer from Allergies Total

Smoker 35 25 60 Nonsmoker 55 185 240 Total 90 210 300 0.50. 0.24. 1.00. 0.00. Explanation

These are mutually exclusive, so the joint probability is zero.

If a fair coin is tossed twice, what is the probability of obtaining heads both times?

✗ B) ✗ C) ✓ D) 3/4. 1/2. 1/4. Explanation

The probability of tossing a head, H, is P(H) = 1/2. Since these are independent events, the probability of two heads in a row is P(H H) = P(H) × P(H) = 1/2 × 1/2 = 1/4.

Topic 16 - Basic Statistics

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In a negatively skewed distribution, what is the order (from lowest value to highest) for the distribution's mode, mean, and median values?

Mean, median, mode.

Median, mode, mean. Mode, mean, median. Median, mean, mode. Explanation

In a negatively skewed distribution, the mean is less than the median, which is less than the mode.

An investor is considering purchasing ACQ. There is a 30% probability that ACQ will be acquired in the next two months. If ACQ is acquired, there is a 40% probability of earning a 30% return on the investment and a 60% probability of earning 25%. If ACQ is not acquired, the expected return is 12%. What is the expected return on this investment?

17.4%. 16.5%. 12.3%. 18.3%. Explanation E(r) = (0.70 × 0.12) + (0.30 × 0.40 × 0.30) + (0.30 × 0.60 × 0.25) = 0.165.

Determine and interpret the correlation coefficient for the two variables X and Y. The standard deviation of X is 0.05, the standard deviation of Y is 0.08, and their covariance is −0.003.

−1.33 and the two variables are negatively associated.

−0.75 and the two variables are negatively associated. +0.75 and the two variables are positively associated. +1.33 and the two variables are positively associated. Explanation

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Which of the following statements about the correlation coefficient is TRUE? The correlation coefficient is:

bounded between 0 and +1.

boundless.

bounded between -1 and 0. bounded between -1 and +1. Explanation

The correlation coefficient is bounded between -1 and +1.

If X and Y are independent random variables:

the variables are perfectly correlated.

The covariance between the two variables is equal to zero and the correlation between the two variables is equal to zero.

the covariance between the two variables is equal to zero. the correlation between the two variables is equal to zero. Explanation

Variables have a zero correlation and covariance when they are independent.

Andrew Dawns holds a large position in the common stock of Savory Doughnuts, Inc (Savory). After an extensive executive search, Savory is about to announce a new CEO. There are three candidates for the CEO position, and each is viewed

differently by the market. Dawns estimates the following probabilities for the rate of return on Savory's stock in the year following the announcement:

Candidate Probability of Being Chosen Rate of Return if Chosen

One

.50

10%

Two

.15

-40%

Three

.35

20%

Based on Dawn's estimates, the expected rate of return on Savory's stock following the announcement of the new CEO is closest to:

6%.

-2%. 9%. 10%.

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✗ A) ✓ B) ✗ C) ✗ D) Explanation E(R) = (0.50 × 10%) + (0.15 × -40%) + (0.35 × 20%) = 6%.

Shawn Choate is thinking about his graduate thesis. Still in the preliminary stage, he wants to choose a variable of study that has the most desirable statistical properties. The statistic he is presently considering has the following characteristics:

The expected value of the sample mean is equal to the population mean.

The variance of the sampling distribution is smaller than that for other estimators of the parameter.

As the sample size increases, the standard error of the sample mean rises and the sampling distribution is centered more closely on the mean.

Select the best choice. Choate's estimator is:

efficient and consistent. unbiased and efficient.

unbiased, efficient, and consistent. unbiased and consistent.

Explanation

The estimator is unbiased because the expected value of the sample mean is equal to the population mean. The estimator is efficient because the variance of the sampling distribution is smaller than that for other estimators of the parameter.

The estimator is not consistent. To be consistent, as the sample size increases, the standard error of the sample mean should fall and the sampling distribution will be centered more closely on the mean. A consistent estimator provides a more accurate estimate of the parameter as the sample size increases.

A security has the following expected returns and probabilities of occurrence:

Return Probability

6.1% 10%

7.5% 40%

9.2% 50%

What is the expected return of the security?

7.6%. 8.2%. 7.8%. 8.4%.

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The expected return is calculated as the sum of the return times the probability that the return occurs. In this case, the expected return is: (0.061)(0.10) + (0.075)(0.40) + (0.092)(0.50) = 0.0821, or 8.2%.

Which of the following statements is least accurate regarding covariance?

Covariance can exceed one.

A covariance of zero rules out any relationship. Covariance can only apply to two variables at a time. Covariance can be negative.

Explanation

A covariance only measures the linear relationship. The covariance can be zero while a non-linear relationship exists. All the other statements are true.

For the case of simple linear regression with one independent variable, which of the following statements about the correlation coefficient is least accurate?

If the regression line is flat and the observations are dispersed uniformly about the line, the correlation coefficient will be +1.

The correlation coefficient describes the strength of the relationship between the X variable and the Y variable.

The correlation coefficient can vary between −1 and +1.

If the correlation coefficient is negative, it indicates that the regression line has a negative slope coefficient.

Explanation

Correlation analysis is a statistical technique used to measure the strength of the relationship between two variables. The measure of this relationship is called the coefficient of correlation.

If the regression line is flat and the observations are dispersed uniformly about the line,there is no linear relationship between the two variables and the correlation coefficient will be zero.

All other choices are TRUE.

SCU and QXA are two stocks in the same industry. The variance of returns for each stock is 0.3025 and the returns are perfectly positively correlated. The covariance between the returns is closest to:

0.3025.

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✗ A) ✗ B) ✓ C) ✗ D) 0.5500. 0.1000. Explanation

Perfectly positively correlated means the correlation coefficient between the two stocks is 1. The standard deviation for both stocks is the square root of 0.3025, or . Therefore:

If Estimator A is a more efficient estimator than Estimator B, it will have:

the same mean and a smaller variance.

the same mean and a larger variance. a smaller mean and the same variance. a smaller mean and a larger variance. Explanation

The more efficient estimator is the one that has the smaller variance, given that they both have the same mean.

The correlation coefficient for two dependent random variables is equal to:

the absolute value of the difference between the means of the two variables divided by the product of the variances.

the product of the standard deviations for the two random variables divided by the covariance. the covariance between the random variables divided by the product of the standard deviations. the covariance between the random variables divided by the product of the variances.

Explanation

Correlation is equal to covariance divided by the product of the standard deviations.

For two (possibly dependent) random variables, X and Y, an upper bound on the covariance of X and Y is:

zero.

1.

σ(X) • σ(Y).

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Cov(X,Y) ≤ σ(X) • σ(Y).

When the tails of a distribution are fatter than that implied by a normal distribution, we say that the distribution is:

leptokurtic.

platykurtic. symmetrical. skewed. Explanation

When the tails of a distribution are thicker than that of a normal distribution, we say that the distribution is leptokurtic. Platykurtic means that the tails are thinner than normal. The size of the tails is not indicative of a distribution's symmetry. Rather, the presence of skewness indicates that the distribution is not symmetrical.

In order to have a negative correlation between two variables, which of the following is most accurate?

The covariance must be negative.

The covariance can never be negative.

Both the covariance and at least one of the standard deviations must be negative. Either the covariance or one of the standard deviations must be negative. Explanation

In order for the correlation between two variables to be negative, the covariance must be negative. (Standard deviations are always positive.)

Use the following joint probability distribution to answer the questions below. Y=1 Y=2 Y=3

X=1 0.05 0.05 0.10 X=2 0.05 0.10 0.15 X=3 0.15 0.15 0.20

The expected value of Y is closest to:

0.2.

2.2. 1.0.

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2.3. Explanation

p(Y=1)=0.05+0.05+0.15=0.25, p(Y=2)=0.05+0.10+0.15=0.30, and p(Y=3)=0.10+0.15+0.20=0.45, so E(Y)=0.25(1)+0.30(2)+0.45(3)=2.20.

If you know that Y is equal to 2, the probability that X is equal to 1 is closest to:

0.20. 0.17. 0.05. 0.25. Explanation p(X=1|Y=2)=0.05/(0.05+0.10+0.15)=0.17.

The variance of X is closest to:

0.74.

0.61. 0.67. 0.54. Explanation

p(X=1)=0.05+0.05+0.10=0.20, p(X=2)=0.05+0.10+0.15=0.30, and p(X=3)=0.15+0.15+0.20=.50. Thus, the mean of X is equal to μ=0.20(1)+0.30(2)+0.50(3)=2.3, and the variance is calculated as 0.20(1-2.3) +0.30(2-2.3) +0.50(3-2.3) =0.61.

A security has the following expected returns and probabilities of occurrence:

Return Probability

10%

40%

12%

40%

15%

20%

What is the expected return of the security?

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✗ A) ✓ B) ✗ C) 11.8%. 12.4%. 11.4%. 12.2%. Explanation

The expected return is calculated as the sum of the return times the probability that the return occurs. In this case, the expected return is: (0.10)(0.40) + (0.12)(0.40) + (0.15)(0.20) = 0.118, or 11.8%.

Y = 1 Y = 2 Y = 3 X = 1 0.05 0.15 0.20 X = 2 0.15 0.15 0.30

The expected value of X is closest to:

1.6 1.2 1.8 1.5 Explanation p(X = 1) = 0.05 + 0.15 + 0.20 = 0.40, and p(X = 2) = 0.15 + 0.15 + 0.30 = 0.60, so E(X) = 0.40(1) + 0.60(2) = 1.6.

If you know that X is equal to 1, the probability that Y is equal to 2 is closest to:

0.30 0.38 0.50 0.15 Explanation p(Y = 2|X = 1) = 0.15/(0.05 + 0.15 + 0.20) = 0.375.

The variance of Y is closest to:

1.51

0.61 0.76

✗ D)

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2.27 Explanation

p(Y = 1) = 0.05 + 0.15 = 0.20, p(Y = 2) = 0.15 + 0.15 = 0.30, and p(Y = 3) = 0.20 + 0.30 = 0.50. Thus, the mean of Y is equal to μ = 0.20(1) + 0.30(2) + 0.50(3) = 2.3, and the variance is calculated as 0.20(1 - 2.3)2 + 0.30(2 - 2.3)2 + 0.50(3 - 2.3)2 = 0.61.

The Y variable is regressed against the X variable resulting in a regression line that is flat with the plot of the paired observations widely dispersed about the regression line. Based on this information, which statement is most accurate?

The R of this regression is close to 100%.

X is perfectly negatively correlated to Y. X is perfectly positively correlated to Y.

The correlation between X and Y is close to zero. Explanation

Perfect correlation means that the observations fall on the regression line. + means the line is going up and − means it is pointed down. An R of 100%, means perfect correlation. When there is no correlation, the regression line is flat and the residual standard error equals the standard deviation of Y.

If the variance of the sampling distribution of an estimator is smaller than all other unbiased estimators of the parameter of interest, the estimator is:

consistent. efficient. unbiased. reliable. Explanation By definition.

The sample mean is an unbiased estimator of the population mean because the:

2

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sampling distribution of the sample mean has the smallest variance of any other unbiased estimators of the population mean.

expected value of the sample mean is equal to the population mean. sampling distribution of the sample mean is normal.

sample mean provides a more accurate estimate of the population mean as the sample size increases.

Explanation

An unbiased estimator is one for which the expected value of the estimator is equal to the parameter you are trying to estimate.

A distribution that has positive excess kurtosis:

is less peaked than a normal distribution. is more peaked than a normal distribution. is more skewed than a normal distribution. has thinner tails than a normal distribution.

Explanation

A distribution with positive excess kurtosis is one that is more peaked than a normal distribution.

Mike Palm, CFA, is an analyst with a large money management firm. Currently, Palm is considering the risk and return parameters associated with Alux, a small technology firm. After in depth analysis of the firm and the economic outlook, Palm estimates the following return probabilities:

Probability (P ) Return (R )

0.3 -5%

0.5 15%

0.2 25%

Palm's objective is to quantify the risk/return relationship for Alux.

Given the returns and probability estimates above, what is the expected return for Alux?

9%. 11%. 15%. 45%. Explanation ER = S(ER )(W ) i i

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ER = w ER + w ER + w ER , where ER = Expected Return and w = % invested in each stock. ER = (0.3 * -5.0) + (0.50 * 15) + (0.20 * 25) = -1.5 + 7.5 + 5.0 = 11.0%

A two-sided but very thick coin is expected to land on its edge twice out of every 100 flips. And the probability of face up (heads) and the probability of face down (tails) are equal. When the coin is flipped, the prize is $1 for heads, $2 for tails, and $50 when the coin lands on its edge. What is the expected value of the prize on a single coin toss?

$1.50.

$26.50. $2.47. $17.67. Explanation

Since the probability of the coin landing on its edge is 0.02, the probability of each of the other two events is 0.49. The expected payoff is: (0.02 × $50) + (0.49 × $1) + (0.49 × $2) = $2.47.

The characteristic function of the product of independent random variables is equal to the:

square root of the product of the individual characteristic functions.

exponential root of the product of the individual characteristic functions. sum of the individual characteristic functions.

product of the individual characteristic functions. Explanation

The characteristic function of the sum of independent random variables is equal to the product of the individual characteristic functions. E(XY) = E(X) × E(Y).

Use the following probability distribution to calculate the standard deviation for the portfolio.

State of the Economy Probability Return on Portfolio

Boom

0.30

15%

Bust

0.70

3%

6.5%.

5.5%.

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7.0%. Explanation

[0.30 × (0.15 − 0.066) + 0.70 × (0.03 − 0.066) ] = 5.5%.

There is a 30% chance that the economy will be good and a 70% chance that it will be bad. If the economy is good, your returns will be 20% and if the economy is bad, your returns will be 10%. What is your expected return?

18%.

15%. 13%. 17%. Explanation

Expected value is the probability weighted average of the possible outcomes of the random variable. The expected return is: ((0.3) × (0.2)) + ((0.7) × (0.1)) = (0.06) + (0.07) = 0.13.

The variance of the sum of two independent random variables is equal to the sum of their variances:

plus zero.

minus a positive covariance term. plus a positive covariance term. plus a non-zero covariance term. Explanation

Independent random variables have a covariance of zero. Hence, the variance of the sum of two variables will be the sum of the variables' variances.

As a fund manager, Bryan Cole, CFA, is responsible for assessing the risk and return parameters of the portfolios he oversees. Cole is currently considering a portfolio consisting of only two stocks. The first stock, Remba Co., has an expected return of 12 percent and a standard deviation of 16 percent. The second stock, Labs, Inc., has an expected return of 18 percent and a standard deviation of 25 percent. The correlation of returns between the two securities is 0.25.

Cole has the option of including a third stock in the portfolio. The third stock, Wimset, Inc., has an expected return of 8% and a standard deviation of 10 percent. If Cole constructed an equally weighted portfolio consisting of all three stocks, the portfolio's expected return would be closest to:

17.1%. 12.7%.

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Question #37 of 54

13.9%. 15.9%.

Explanation

ER = S(ER )(W )

ER = w ER + w ER + w ER , where ER = Expected Return and w = % invested in each stock.

Here, use 1/3 for each of the weightings. (Note: If you use 0.33, you will calculate a slightly different result.)

ER =( 1/3 * 12) + (1/3 * 18) + (1/3 * 8) = 4 + 6 + 2.7 = 12.7%

A scatter plot is a collection of points on a graph where each point represents the values of two variables (i.e., an X/Y pair). The pattern of data points will illustrate a correlation between these two variables that is between:

-0.7 and +0.7.

-1 to +1.

slope of -1 and slope of +1. 0 and +1.

Explanation

Correlations range from - 1 to + 1. For r = 1 and r = -1 the data points lie exactly on a line, but the slope of that line is not necessarily +1 or -1.

A bond analyst is looking at historical returns for two bonds, Bond 1 and Bond 2. Bond 2's returns are much more volatile than Bond 1. The variance of returns for Bond 1 is 0.012 and the variance of returns of Bond 2 is 0.308. The correlation between the returns of the two bonds is 0.79, and the covariance is 0.048. If the variance of Bond 1 increased to 0.026 while the variance of Bond B decreased to 0.188 and the covariance remains the same, the correlation between the two bonds will:

increase.

decrease.

the values given are not plausible. remain the same.

Explanation

P = 0.048/(0.026 × 0.188 ) = 0.69 which is lower than the original 0.79.

Suppose the covariance between Y and X is 12, the variance of Y is 25, and the variance of X is 36. What is the correlation coefficient (r), between Y and X?

portfolio stock % of funds invested in each of the stocks

1 1 2 2 3 3

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Question #39 of 54

✗ A) ✗ B) 0.013. 0.400. 0.160. 0.000. Explanation

The correlation coefficient is:

An analyst is currently considering a portfolio consisting of two stocks. The first stock, Remba Co., has an expected return of 12% and a standard deviation of 16%. The second stock, Labs, Inc., has an expected return of 18% and a standard deviation of 25%. The correlation of returns between the two securities is 0.25.

If the analyst forms a portfolio with 30% in Remba and 70% in Labs, what is the portfolio's expected return?

16.2%. 15.0%. 17.3%. 21.5%. Explanation ER = Σ(ER )(W )

ER = w ER + w ER , where ER = Expected Return and w = % invested in each stock. ER = (0.3 × 12) + (0.70 × 18) = 3.6 + 12.6 = 16.2%

An investor owns the following three-stock portfolio.

Stock Market Value Expected Return

A $5,000 12%

B $3,000 8%

C $4,000 9%

The expected return is closest to:

9.67%. 29.00%.

portfolio stock % of funds invested in each of the stocks

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Question #40 of 54

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Question #41 of 54

10.50%. 10.00%.

Explanation

To calculate this result, we first need to calculate the portfolio value, then determine the weights for each stock, and then calculate the expected return.

Portfolio Value: = sum of market values = 5,000 + 3,000 + 4,000 = 12,000

Portfolio Weights: W = 5,000 / 12,000 = 0.4167 W = 3,000 / 12,000 = 0.2500 W = 4,000 / 12,000 = 0.333 Expected Return ER = S(ER )(W )

ER = w ER + w ER + w ER , where ER = Expected Return and w = % invested in each stock. ER = (0.4167 * 12.0) + (0.2500 * 8.0) + (0.333 * 9.0) = 10.0%

Tully Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Tully's economist has estimated the probability of each scenario as shown in the table below. Given this information, what is the expected return on portfolio A?

Scenario Probability Return on Portfolio A Return on Portfolio B

A

15%

17%

19%

B

20%

14%

18%

C

25%

12%

10%

D

40%

8%

9%

The expected return is equal to the sum of the products of the probabilities of the scenarios and their respective returns: = (0.15) (0.17) + (0.20)(0.14) + (0.25)(0.12) + (0.40)(0.08) = 0.1155 or 11.55%.

A B C

portfolio stock % of funds invested in each of the stocks

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Question #43 of 54

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Suppose you have chosen two stocks with an equity investment screener. The following data has been collected on these two securities:

Stock A Stock B

Expected Return 5% 8%

Variance 10% 16%

Covariance (A,B) 0.2

What is the mean and variance of the sum of the two stock's expected returns and variances assuming the returns are independent of one another? Mean = 6.5%; Variance = 13.4%. Mean = 13%; Variance = 26.4%. Mean = 6.5%; Variance = 13.0%. Mean = 13%; Variance = 26.0%. Explanation

The mean of the sum of random variables is computed as: E(A +B) = E(A) + E(B) = 5% + 8% = 13%. The variance of the sum of random variables is computed as: Var(A + B) = Var(A) + Var(B) = 10% + 16% = 26%. Since the returns were assumed to be independent, the covariance was not a factor in the variance calculation.

An analyst has knowledge of the beginning-of-period expected returns, standard deviations of return, and market value weights for the assets that comprise a portfolio. The analyst does not require the covariances of returns between asset pairs to calculate the:

variance of the return on the portfolio.

reduction in risk due to diversification. correlations between asset pairs. expected return on the portfolio. Explanation

All that is required to calculate the expected return for the portfolio is the portfolio weights and individual asset expected returns. All other items are functions of the covariance.

Rafael Garza, CFA, is considering the purchase of ABC stock for a client's portfolio. His analysis includes calculating the covariance between the returns of ABC stock and the equity market index. Which of the following statements regarding Garza's analysis is most accurate?

The covariance of two variables is an easier measure to interpret than the correlation coefficient.

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Question #45 of 54

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The actual value of the covariance is not very meaningful because the measurement is very sensitive to the scale of the two variables.

The covariance measures the strength of the linear relationship between two variables. Explanation

Covariance is a statistical measure of the linear relationship of two random variables, but the actual value is not meaningful because the measure is extremely sensitive to the scale of the two variables. Covariance can range from negative to positive infinity.

Left-skewed distributions exhibit:

greater mass to the right of the expected value.

greater mass close to the expected value. greater mass to the left of the expected value. a longer tail to the right of the distribution. Explanation

Left-skewed distributions are those with a longer tail to the left side of the distribution, and whose mass is to the right of the expected value.

Tully Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Tully's economist has estimated the probability of each scenario, as shown in the table below. Given this information, what is the standard deviation of returns on portfolio A?

Scenario Probability Return on Portfolio A Return on Portfolio B

A

15%

18%

19%

B

20%

17%

18%

C

25%

11%

10%

D

40%

7%

9%

Question #46 of 54

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Question #47 of 54

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Question #48 of 54

✗ A) ✓ B) ✗ C) ✗ D) σ = 0.0452836

Consider the case when the Y variable is in U.S. dollars and the X variable is in U.S. dollars. The 'units' of the covariance between Y and X are:

U.S. dollars.

squared U.S. dollars.

a range of values from −1 to +1. the square root of U.S. dollars. Explanation

The covariance is in terms of the product of the units of Y and X. It is defined as the average value of the product of the deviations of observations of two variables from their means. The correlation coefficient is a standardized version of the covariance, ranges from −1 to +1, and is much easier to interpret than the covariance.

Thomas Manx is attempting to determine the correlation between the number of times a stock quote is requested on his firm's website and the number of trades his firm actually processes. He has examined samples from several days trading and quotes and has determined that the covariance between these two variables is 88.6, the standard deviation of the number of quotes is 18, and the standard deviation of the number of trades processed is 14. Based on Manx's sample, what is the correlation between the number of quotes requested and the number of trades processed?

0.78. 0.35. 0.18. 0.98.

Explanation

Correlation = Cov (X,Y) / (Std. Dev. X)(Std. Dev. Y) Correlation = 88.6 / (18)(14) = 0.35

The covariance of returns on two investments over a 10-year period is 0.009. If the variance of returns for investment A is 0.020 and the variance of returns for investment B is 0.033, what is the correlation coefficient for the returns?

0.687.

0.350. 0.444. 0.785. Explanation

Question #49 of 54

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Question #50 of 54

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Question #51 of 54

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The correlation coefficient is: Cov(A,B) / [(Std Dev A)(Std Dev B)] = 0.009 / [(√0.02)(√0.033)] = 0.350.

The sample mean is a consistent estimator of the population mean because the:

sample mean provides a more accurate estimate of the population mean as the sample size increases.

sampling distribution of the sample mean has the smallest variance of any other unbiased estimators of the population mean.

sampling distribution of the sample mean is normal.

expected value of the sample mean is equal to the population mean. Explanation

A consistent estimator provides a more accurate estimate of the parameter as the sample size increases.

In a positively skewed distribution, what is the order (from lowest value to highest) for the distribution's mode, mean, and median values?

Mode, mean, median.

Mode, median, mean. Mean, median, mode. Median, mean, mode. Explanation

In a positively skewed distribution, the mode is less than the median, which is less than the mean.

An investor owns the following portfolio today.

Stock Market Value Expected Annual Return

R $2,000 17%

S $3,200 8%

T $2,800 13%

The investor's expected total rate of return (increase in market value) after three years is closest to:

40.5%.

136.0%. 36.0%. 12.0%.

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Question #53 of 54

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Question #54 of 54

Explanation

To calculate this result, we first need to calculate the portfolio value, then determine the weights for each stock, and then calculate the expected return. Finally, we determine the compounded rate after three years.

Portfolio Value: = sum of market values = 2,000 + 3,200 + 2,800 = 8,000 Portfolio Weights: W = 2,000 / 8,000 = 0.25 W = 3,200 / 8,000 = 0.40 W = 2,800 / 8,000 = 0.35 Expected Return ER = Σ[(ER )(W )]

ER = w ER + w ER + w ER , where ER = Expected Return and w = % invested in each stock. ER = (0.25 × 17.0) + (0.40 × 8.0) + (0.35 × 13.0) = 12.0%

Expected Return after three years

= (1 + return) = (1.12) − 1 = 1.405 − 1 = 0.405, or 40.5%.

It is often said that stock returns are leptokurtic. If this is true, relative to a normal distribution of the same mean and variance, the distribution of stock returns is:

thin-tailed.

negatively skewed. fat-tailed.

positively skewed. Explanation

A leptokurtic distribution is more peaked and has fatter tails than a normal distribution.

Which model does not lend itself to correlation coefficient analysis?

Y = X . Y - X = 2. X = Y × 2. Y = X + 2.

Explanation

The correlation coefficient is a measure of linear association. All of the functions except for Y = X are linear functions. Notice that Y - X = 2 is the same as Y = X + 2.

A B C

portfolio stock % of funds invested in each of the stocks

R R S S T T

3 3

3

✗ A) ✗ B) ✗ C) ✓ D) The covariance: must be positive.

must be between -1 and +1. must be less than +1. can be positive or negative.

Explanation

Topic 17 - Distributions

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Standardizing a normally distributed random variable requires the:

mean and the standard deviation.

variance and kurtosis. natural logarithm of X.

mean, variance and skewness. Explanation

All that is necessary is to know the mean and the variance. Subtracting the mean from the random variable and dividing the difference by the standard deviation standardizes the variable.

Which of the following statements is true regarding the mixture of distributions for risk modeling? I. Distributions should never be combined.

II. It may be helpful to create a new distribution if the underlying data you are working with does not currently fit a pre-determined distribution.

I only.

II only. Neither I nor II. Both I and II. Explanation

Distributions can be combined to create unique probability density functions. It may be helpful to create a new distribution if the underlying data you are working with does not currently fit a pre-determined distribution. In this case, a newly created distribution may assist with explaining the relevant data set.

Which of the following statements about a normal distribution is least accurate?

A normal distribution has excess kurtosis of three.

The mean, median, and mode are equal.

Approximately 68% of the observations lie within +/- 1 standard deviation of the mean. The mean and variance completely define a normal distribution.

Explanation

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distributions is three; an excess kurtosis of three would indicate a leptokurtic distribution. The other choices are true.

The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution:

becomes smaller.

approaches the mean. becomes larger.

approaches a normal distribution. Explanation

As n gets larger, the variance of the distribution of sample means is reduced, and the distribution of sample means approximates a normal distribution.

Given the probabilities N(-0.5) = 0.3085, N(0.75) = 0.7734, and N(1.50) = 0.9332 from a z-table, the probability of 0.2266 corresponds to: N(0.50). N(-0.75). N(0.25). N(-0.25). Explanation

This problem is checking your knowledge of a normal distribution and gives you more information than you need to answer the question. The area of a normal distribution is 1, with two symmetric halves that equal 0.5 each. N(0.75) means that the area to the left of 0.75 on the positive portion of the curve is 0.7734. This means that the area to the right of 0.75 is (1.0 - 0.7734) = 0.2266. Since the halves of a normal distribution are symmetrical, that means the area to the left of (-0.75) is also 0.2266.

An analyst has determined that the probability that the S&P 500 index will increase on any given day is 0.60 and the probability that it will decrease is 0.40. The expected value and variance of the number of up days in a 5-day period are closest to:

2.0 and 2.1. 3.0 and 1.2. 3.0 and 1.1. 2.0 and 0.5. Explanation E(X)=np=5(0.60)=3.0. Var(X)=np(1-p)=5(0.60)(0.40)=1.2.

Question #7 of 32

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Question #8 of 32

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Question #9 of 32

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A normal distribution can be completely described by its:

standard deviation. skewness and kurtosis. mean and mode. mean and variance.

Explanation

The normal distribution can be completely described by its mean and variance.

Which of the following statements about probability distributions is least accurate?

A probability distribution is, by definition, normally distributed.

A probability distribution includes a listing of all the possible outcomes of an experiment. In a binomial distribution each observation has only two possible outcomes that are mutually exclusive.

One of the key properties of a probability function is 0 ≤ p ≤ 1. Explanation

Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures.

The distribution of annual returns for a bond portfolio is approximately normal with an expected value of $120 million and a standard deviation of $20 million. Which of the following is closest to the probability that the value of the portfolio one year from today will be between $110 million and $170 million?

66%.

58%. 74%. 42%. Explanation

Calculating z-values, z = (110 − 120) / 20 = −0.5. z = (170 − 120) / 20 = 2.5. Using the z-table, P(−0.5) = (1 − 0.6915) = 0.3085. P(2.5) = 0.9938. P(−0.5 < X < 2.5) = 0.9938 − 0.3085 = 0.6853. Note that on the exam, you will not have access to z-tables, so you would have to reason this one out using the normal distribution approximations. You know that the probability within +/− 1 standard deviation of the mean is approximately 68%, meaning that the area within −1 standard deviation of the mean is 34%. Since −0.5 is half of −1, the area under −0.5 to 0 standard deviations under the mean is approximately 34% / 2 = 17%. The probability under +/− 2.5 standard deviations of the mean is approximately 99%. The value $170 is mid way between +2 and +3 standard deviations, so the probability between these values must be (99% / 2) = 2%. The value from 0 to 2.5

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standard deviations must therefore be (99% / 2) − (2% / 2) = 48.5%. Adding these values gives us an approximate probability of (48.5% + 17%) = 65.5%.

The central limit theorem concerns the sampling distribution of the:

sample standard deviation.

population mean.

population standard deviation. sample mean.

Explanation

The central limit theorem tells us that for a population with a mean m and a finite variance σ , the sampling distribution of the sample means of all possible samples of size n will approach a normal distribution with a mean equal to m and a variance equal to σ / n as n gets large.

Which one of the following statements about the t-distribution is most accurate?

The t-distribution approaches the standard normal distribution as the number of degrees of freedom becomes large.

Compared to the normal distribution, the t-distribution is flatter with less area under the tails. Compared to the normal distribution, the t-distribution is more peaked with more area under the tails. The t-distribution is the appropriate distribution to use when constructing confidence intervals based on large samples.

Explanation

As the number of degrees of freedom grows, the t-distribution approaches the shape of the standard normal distribution. Compared to the normal distribution, the t-distribution is less peaked with more area under the tails. When choosing a distribution, three factors must be considered: sample size, whether population variance is known, and if the distribution is normal.

For a normal distribution, what approximate percentage of the observations falls within ± 2 standard deviations of the mean?

92%. 90%. 95%. 99%. Explanation 2 2

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For normal distributions, approximately 95 percent of the observations fall within ± 2 standard deviations of the mean.

Regarding a binomially distributed random variable, the variance of the number of successes, X, where n equals the number of trials and p equals the probability of success, is most likely equal to:

np.

p. p(1-p). np(1 - p). Explanation

For a given series of n trials, the variance of X for a binomial random variable is equal to: np(1 - p) = npq, where q is the probability of failure.

An investment has a mean return of 15% and a standard deviation of returns equal to 10%. Which of the following statements is least accurate? The probability of obtaining a return:

greater than 35% is 0.025. greater than 25% is 0.32. between 5% and 25% is 0.68. less than 5% is 0.16.

Explanation

Sixty-eight percent of all observations fall within +/- one standard deviation of the mean of a normal distribution. Given a mean of 15 and a standard deviation of 10, the probability of having an actual observation fall within one standard deviation, between 5 and 25, is 68%. The probability of an observation greater than 25 is half of the remaining 32%, or 16%. This is the same probability as an observation less than 5. Because 95% of all observations will fall within 20 of the mean, the probability of an actual observation being greater than 35 is half of the remaining 5%, or 2.5%.

The return on a portfolio is normally distributed with a mean return of 8 percent and a standard deviation of 18 percent. Which of the following is closest to the probability that the return on the portfolio will be between -27.3 percent and 37.7 percent?

92.5%.

68.0%. 81.5%. 96.5%. Explanation

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Note that if you memorize the basic intervals for a normal distribution, you do not need a normal distribution table to answer this question. -27.3% represents a loss of -35.3% from the mean return (-27.3 - 8.0), which is (-35.3/18) = -1.96 standard deviations to the left of the mean. For a normal distribution, we know that approximately 95 percent of all observations lie with +/- 1.96 standard deviations of the mean, so the probability that the return is between -27.3% and 8.0% must be (95%/2) = 47.5%. A return of 37.7 percent represents a gain of (37.7 - 8.0) = 29.7% from the mean return, which is (29.7/18) = 1.65 standard deviations to the right of the mean. For a normal distribution, we know that approximately 90 percent of all observations lie with +/- 1.65 standard deviations of the mean, so the probability that the return is between 8.0% and 37.7% must be (90%/2) = 45%. Therefore the probability that the return is between -27.3 percent and 37.7 percent = (47.5% + 45%) = 92.5%.

The lower limit of a normal distribution is:

negative one.

zero. one.

negative infinity. Explanation

By definition, a true normal distribution has a positive probability density function from negative to positive infinity.

With 60 observations, what is the appropriate number of degrees of freedom to use when carrying out a statistical test on the mean of a population? 59. 60. 58. 61. Explanation

When performing a statistical test on the mean of a population based on a sample of size n, the number of degrees of freedom is n - 1 since once the mean is estimated from a sample there are only n - 1 observations that are free to vary. In this case the appropriate number of degrees of freedom to use is 60 - 1 = 59.

Approximately 50 percent of all observations for a normally distributed random variable fall in the interval:

µ ± 2σ

µ ± 0.67σ µ ± σ µ ± 3σ

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Question #21 of 32

✓ A) ✗ B) ✗ C) ✗ D) Explanation µ ± 0.67σ

The annual returns for a portfolio are normally distributed with an expected value of £50 million and a standard deviation of £25 million. What is the probability that the value of the portfolio one year from today will be between £91.13 million and £108.25 million? 0.075. 0.090. 0.040. 0.025. Explanation

Calculate the standardized variable corresponding to the outcomes. Z = (91.13 − 50) / 25 = 1.645, and Z = (108.25 − 50) / 25 = 2.33. The cumulative normal distribution gives cumulative probabilities of F(1.645) = 0.95 and F(2.33) = 0.99. The probability that the outcome will lie between Z and Z is the difference: 0.99 − 0.95 = 0.04. Note that even though you will not have a z-table on the exam, the probability values for 1.645 and 2.33 are commonly used values that you should have memorized.

When is the t-distribution the appropriate distribution to use? The t-distribution is the appropriate distribution to use when constructing confidence intervals based on:

small samples from populations with known variance that are at least approximately normal.

large samples from populations with known variance that are nonnormal.

small samples from populations with unknown variance that are at least approximately normal. large samples from populations with known variance that are at least approximately normal. Explanation

The t-distribution is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance that are either normal or approximately normal.

To apply the central limit theorem to the sampling distribution of the sample mean, the sample is usually considered to be large if n is greater than: 30. 25. 15. 20. 1 2 1 2

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Question #24 of 32

Explanation

A sample size is considered sufficiently large if it is larger than or equal to 30.

Which of the following statements is incorrect regarding a Bernoulli distributed random variable. I. It only has three possible outcomes.

II. It is commonly used for assessing whether or not a company defaults during a specified time period.

Neither I nor II.

I only. Both I and II. II only. Explanation

Bernoulli distributed random variable only has two possible outcomes. The outcomes can be defined as either a "success" or a "failure."

A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account?

1.000.

0.950. 0.977. 0.750. Explanation

Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ − 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 − 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.

A group of investors wants to be sure to always earn at least a 5% rate of return on their investments. They are looking at an investment that has a normally distributed probability distribution with an expected rate of return of 10% and a standard deviation of 5%. The probability of meeting or exceeding the investors' desired return in any given year is closest to:

✗ B) ✓ C) ✗ D)

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Question #27 of 32

✗ A)

98%. 84%. 50%.

Explanation

The mean is 10% and the standard deviation is 5%. You want to know the probability of a return 5% or better. 10% - 5% = 5% , so 5% is one standard deviation less than the mean. Thirty-four percent of the observations are between the mean and one standard deviation on the down side. Fifty percent of the observations are greater than the mean. So the probability of a return 5% or higher is 34% + 50% = 84%.

Which of the following statements about the normal probability distribution is most accurate?

Sixty-eight percent of the area under the normal curve falls between 0 and +1 standard deviations above the mean.

The standardized normal distribution has a mean of zero and a standard deviation of 10.

Five percent of the normal curve probability falls more than outside two standard deviations from the mean.

The normal curve is asymmetrical about its mean. Explanation

The normal curve is symmetrical with a mean of zero and a standard deviation of 1 with 34% of the area under the normal curve falling between 0 and +1 standard deviation above the mean. Ninety-five percent of the normal curve is within two standard deviations of the mean, so five percent of the normal curve falls outside two standard deviations from the mean.

Which one of the following distributions is described entirely by the degrees of freedom?

Lognormal distribution.

Student's t-distribution. Binomial distribution. Normal distribution. Explanation

Student's t-distribution is defined by a single parameter known as the degrees of freedom.

A call center receives an average of two phone calls per hour. The probability that they will receive 20 calls in an 8-hour day is closest to:

✗ B) ✗ C) ✓ D)

Question #28 of 32

✓ A) ✗ B) ✗ C) ✗ D)

Question #29 of 32

✗ A) ✓ B) ✗ C) ✗ D)

Questions #30-31 of 32

Question #30 of 32

16.56%. 6.40%. 5.59%. Explanation

To solve this question, we first need to realize that the expected number of phone calls in an 8-hour day is λ = 2 × 8 = 16. Using the Poisson distribution, we solve for the probability that X will be 20.

P(X = x) = (λ e ) / x!

P(X = 20) = (16 e ) / 20! = 0.0559 = 5.59%

The probability of returns less than −10%, assuming a normal distribution with expected return of 6.5% and standard deviation of 10%, is:

approximately 5%.

not defined with only this information. approximately 10%.

less than 2.5%. Explanation

−10 is 16.5% below the mean return, (16.5 / 10) = 1.65 standard deviations, which leaves approximately 5% of the possible outcomes in the left tail below −10%.

Which statement best describes the properties of Student's t-distribution? The t-distribution is:

symmetrical, defined by two parameters and is less peaked than the normal distribution.

symmetrical, defined by a single parameter and is less peaked than the normal distribution. skewed, defined by a single parameter and is more peaked than the normal distribution. skewed, defined by a single parameter and is less peaked than the normal distribution. Explanation

The t-distribution is symmetrical like the normal distribution but unlike the normal distribution is defined by a single parameter known as the degrees of freedom and is less peaked than the normal distribution.

A study of hedge fund investors found that their household incomes are normally distributed with a mean of $280,000 and a standard deviation of $40,000.

x -λ

✗ A) ✗ B) ✓ C) ✗ D)

Question #31 of 32

✗ A) ✗ B) ✗ C) ✓ D)

Question #32 of 32

✓ A) ✗ B) ✗ C) ✗ D)

The percentage of hedge fund investors that have incomes greater than $350,000 is closest to:

5.0%.

3.0%. 4.0%. 25.0%. Explanation

z = (350,000 280,000)/40,000 = 1.75. Using the ztable, F(1.75) = 0.9599. So, the percentage greater than $350,000 = (1 -0.9599) = 4.0%.

The percentage of hedge fund investors with income less than $180,000 is closest to:

6.48%.

1.15%. 2.50%. 0.62%. Explanation

z = (180,000 - 280,000)/40,000 = -2.50. Using the z-table, F(-2.50) = (1 - 0.9938) = 0.62%.

A food retailer has determined that the mean household income of her customers is $47,500 with a standard deviation of $12,500. She is trying to justify carrying a line of luxury food items that would appeal to households with incomes greater than $60,000. Based on her information and assuming that household incomes are normally distributed, what percentage of households in her customer base has incomes of $60,000 or more? 15.87%. 34.13%. 2.50%. 5.00%. Explanation Z = ($60,000 - $47,500) / $12,500 = 1.0

From the table of areas under the normal curve, 84.13% of observations lie to the left of +1 standard deviation of the mean. So, 100% -84.13% = 15.87% with incomes of $60,000 or more.