RandomVariables.pdf

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www.MasterMathMentor.com - 1 - Stu Schwartz

Random Variables - Terms and Formulas

Random Variable – a variable whose value is a numerical outcome of a random phenomenon.

Example 1: Put all the letters of the alphabet in a hat. If you choose a consonant, I pay you $1. If you choose a vowel, I pay you $5. X is the random variable representing the outcome of the experiment. Its possible values are 1 and 5.

Example 2: In most college courses, you get as a grade, either an A, B, C, D, or F. For credit purposes, A’s are given 4 points, B’s are given 3 points, C’s are given 2 points, Ds are given 1 point, and F’s are no points. Let X

be the random variable representing the points a student gets. The possible values of X are 4, 3, 2, 1, and 0.

Discrete Random Variables have a countable number of possible values. The probability distribution of X lists all possible values of X and their probabilities:

Value of X x1 x2 x3 … xn

Probability of X p₁ p₂ p₃ … p_n

The probabilities must satisfy two requirements:

Every probability pi is a number between 0 and 1.

p1+ p2+p3+...+pn =1

Example 3: Put all the letters of the alphabet in a hat. If you choose a consonant, I pay you $1. If you choose a vowel, I pay you $5. X is the random variable representing the outcome of the experiment. Create the

distribution of X.

Example 4: A college instructor teaching a large class traditionally gives 10% A’s, 20% B’s, 45% C’s, 15% D’s, and 10% F’s. If a student is chosen at random from the class, the student’s grade on a 4-point scale (A = 4) is a random variable X. Create the distribution of X.

What is the probability that a student has a grade point of 3 or better in this class?

What is the probability that a student has a grade point of 2 or worse in this class?

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Continuous Random Variables take on all possible values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that makes up that area.

Example 5: A random number is chosen from 0 to 1. Answer the following questions:

a. Find P(X > .5) b. Find P(X ≥ .5) c. Find P(X ≤ .8) d. Find P(X ≥ .15)

e. Find P(.2 <X < .4) f. Find P(.2 <X ≤ .4) g. Find P(X < 1) h. Find P(X = .5)

Mean of a Random Variable – if an experiment with random variable X is done over a long period of time, we can calculate the mean (average) value of that random variable. Another term for mean of a random variable X

is the expected value of X.

If X is a discrete random variable whose distribution is given by

Probability of X p₁ p₂ p₃ … p_n

To find the mean (or expected value) of X, multiply every possible value of X by its probability Then add the results. The symbol for the mean of X is µX.

µX =x1p1+x2p2+x3p3+ ... +xnpn =

∑

xipi

Example 6: A fair coin is flipped 3 times. Find the mean of the discrete random variable X that counts the number of heads.

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Law of Large Numbers: Draw independent observations from any population with finite mean µ. As the number of observations increases, the mean x of the observed values eventually approaches the mean µ of that

population.

Example 8: If I flip a fair coin, over a long period of time, the percentage of heads will approach 50%. That does not mean that the number of heads will approach the number of tails. Here is a typical experiment in tossing coins.

Heads 7 42 463 4,875 49,660

Tails 3 58 537 5,125 50,340

Percentage of heads 70% 42% 46.3% 48.75% 49.66%

Example 9: If you play blackjack perfectly at a casino, the expected value for every dollar played is 98.5 cents. That means on the average of hand you play, you lose 1.5 cents. Can an individual player win playing blackjack at a casino. The “law of small numbers” says that over a short run, a player can certainly win. But over the long run, the law of large numbers says that a player will eventually lose. And given the huge number of players who play daily, the law of large numbers states that the casino has to win. There is a reason that casinos look the way they do – it is guaranteed money. Still, players can win over the short haul. The question that crops up is what is considered “large?” There is no answer to that. What you have to know is that the larger n (the number of trials) gets, the closer the average win (or loss) per play approaches the true average of losing 1.5 cents a play.

Rules for Means

Rule 1: If X and Y are random variables, then µ_X₊_Y =µX +µY

Rule 2: If X is a random variable and a and b are constants, thenµa+bX =a+bµX

Example 10: You play two casino games. Game 1 has an expectation of losing $1 a play and game 2 has an expectation of losing $2 a play. If you play both games, what is your expectation for both games.

Example 11: You go to a casino and play the same slot machine which averages losing 15 cents a play. You play the machine 100 times and then leave, paying $5 for parking. Find your expectation for your casino visit.

Example 12: Depending on the attendance of a minor league baseball team, the number of hot dogs and sodas sold at a game is given by the following table.

Hot dogs sold 100 200 500 1,000 2,500 Sodas sold 100 200 400 800 1,500 Probability .05 .1 .3 .35 .2 Probability .1 .15 .25 .3 .2

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Variance and standard deviation of a Discrete Random Variable

If X is a discrete random variable whose distribution is given by

Probability of X p1 p2 p3 … pn

We know that the mean (expected value) of X

( )

µX is given by

µX =x1p1+x2p2+x3p3+ ... +xnpn =

∑

xipi

( )

σ_X , we know that σX

( )

= σ2

X

Example 13: A fair coin is flipped 3 times. Find the standard deviation of the discrete random variable X that counts the number of heads.

Example 14: The daily lottery costs $1 to play. You pick a 3 digit number. If you win, you win $500. Find the standard deviation of the lottery.

Example 15: Choose an American household at random and let the random variable X be the number of persons living in the household. Find the standard deviation of the average American household.

X 1 2 3 4 5 6 7 or more

Probability .25 .32 .17 .15 .07 .03 .01

When a problem has many values of X like this one, this can be done using your list feature of your calculator:

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2. To get µX, use L3 to multiply L1 and L2.

3. To find µX, find the sum of L3.

4. We no longer need L3. In L3, we will put our formula for variance:

5. To find the variance σ2

X, find the sum of L3. The standard deviation is the square root.

Rules for Variances of Random Variables:

Rule 1: If X is a random variable and a and b are constants, then

Variance Standard Deviation

σ2

a+bX =b2σ2X σ_a_+bX =bσ_X

Rule 2: If X and Y are independent random variables, then

Variance Standard Deviation

σ2

X+Y =σ2X +σ2Y σX₊_Y = σ2X +σ2Y (Note that you don’t add standard deviations)

σ2

X−Y =σ2X +σ2Y σX_−Y = σ2X +σ2Y

(Note that even though you are finding the variance of the difference between X and Y, you add the variances).

Example 16: Suppose two pro bowlers Adam (A) and Bart (B) have the following distribution of scores.

µA =209 σA =14 µB =221 σA =22

Find the following.

a) µB+A b) µB−A c) 3σA d) σB+A e) σB−A

Example 17: Depending on the attendance of a minor league baseball team, the number of hot dogs and sodas sold at a game is given by the following table.

Hot dogs sold 100 200 500 1,000 2,500 Sodas sold 100 200 400 800 1,500 Probability .05 .1 .3 .35 .2 Probability .1 .15 .25 .3 .2

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Discrete Random Variables

1. Choose an American household at random and let the random variable X be the number of persons living in the household. The probability of X is:

X 1 2 3 4 5 6 7 or more

Probability .25 .32 .17 .15 .07 .03 .01

a. Find P(X > 4) b. Find P(X ≥ 4) c. Find P(2 <X ≤ 4) d. Find P(X ≠ 2)

2. A fair coin is flipped 4 times. Find the probability distribution of the discrete random variable X that counts the number of heads. Then draw the probability histogram for X.

a. Find P(X > 2) b. Find P(X ≥ 2) c. Find P(X ≥ 1) d. Find P(X ≥ 0)

3. A coin that is rigged is flipped 3 times. The probability of heads is twice the probability of tails. Find the probability distribution of the discrete random variable X that counts the number of heads. Then draw the probability histogram for X.

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4. The numbers 1, 2, 3, 4, 5 are placed in a hat. A number is chosen at random, replaced, and another number is chosen at random. Let X be the number of odd numbers that are chosen. Find the probability distribution of the random variable X and draw a probability histogram.

a. Find P(X > 1) b. Find P(X ≥ 1) c. Find P(X ≤ 1) d. Find P(X ≠ 2)

5. An SRS of 3 students are chosen to be on a committee that advises a university on its computer policies. 70% of the student population are PC users and 30% are Mac users. Let X be the number of PC users on the student committee. Find the probability distribution of the random variable X and draw a probability

histogram.

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6. A boss of a company chooses 4 workers at random to act as advisers to him. The company is 75% white and 25% African American. Let X be the number of African-Americans on the committee. Find the probability distribution of the random variable X and draw a probability histogram.

a. Find P(X > 2) b. Find P(X < 2) c. Find P(1 ≤X ≤ 3) d. Find P(X ≠ 0)

7. A density curve is shown on the figure on the right. Find the following probabilities.

a. Find P(X > 1) b. Find P(X < 1.5) c. Find P(.5 ≤X ≤ 1.5) d. Find P(X <2)

8. Two random numbers from 0 to 1 are chosen and added. Let X be their sum. X is thus a continuous random variable between 0 and 2. The density curve is shown to the right. Find the following probabilities:

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Mean of a Random Variable (Expected Value) Problems

1) Choose an American suburban household at random and let the random variable X be the number of cars that people in the house own. Find the number of cars owned by the average suburban American household.

X 0 1 2 3 4 5 6 or more

Probability .08 .29 .32 .17 .08 .04 .02

2) A huge cookie jar has 60% chocolate cookies and 40% vanilla cookies. Sam chooses 3 cookies blindfolded. Let X be the number of chocolate cookies he chooses. Construct the probability distribution of X and the average number of chocolate cookies he chooses.

3) A contractor is bidding on a road construction job that promises a profit of $20,000 with a probability of ₁₀7 and a loss, due to strikes, weather conditions, late arrival of building materials, and so on, of $40,000 with a probability of ₁₀3 . What is the contractor’s mean expectation?

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5) A box contains 8 green marbles, 7 yellow marbles, and 5 black marbles. One marble is to be selected from the box. You get $10 if the marble selected is black, but lose $3 if the marble is green and lose $5 if the marble is yellow. What is your mean expectation?

6) A coin is tossed three times. If heads appears on all 3 tosses, Mary will win $16. If heads appears on 2 of the tosses, she will win $2. The game costs $5 to play. What is her mean expectation?

7) A carnival has set up the following game. You are blindfolded and have to pick a coin from 9 pennies, 8 nickels, 12 dimes, 16 quarters, 11 half-dollars, and 4 dollar pieces. It costs a quarter to play the game. What could you be expected to win or lose every time you play the game?

8) A pair of dice is rolled. Depending on the sum of the dots on both dice, Jake can win or lose money as shown in the following chart. What is Jake’s mean expectation?

Sum of Dots Outcome 6 or 8 or 9 win $6 2 or 4 or 5 lose $4

10 win $7

3 or 12 win $5

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9) An oil company will only invest in an oil well if the can expect to make at least $1 million in profit. They find a possible area in which to drill in Canada. It will cost the company $3 million to make the attempt. If they find oil, they can expect to clear $7 million in profits. Geologists have estimated that the probability of striking oil is 0.35. Should the company make the attempt and why or why not?

10) A wheel of fortune below costs $2 to play. If the spinner stops on black or blue after one spin, the prize is $5. If it stops on green, the prize is $3. If it stops on yellow, the prize is $0.50. If it stops on brown or red, there is no payoff. What is your mean expectation every time you spin the wheel. Should you play the game?

11) In a raffle costing $1 a ticket, 225 tickets are sold. First place gets $50. 2 second place tickets gets $30 and 3 third place tickets gets $20. What is the mean value of your return on a ticket?

12) An ecologist collected the data shown in the table below on the life span of a certain species of deer. Based on this sample, what is the expected lifespan of this species?

Age at death (years) 1 2 3 4 5 6 7 8

Number 2 30 86 132 173 77 40 10

Black Green

Yellow Red

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13) A drug is administered to sets of three patients. Over a period of time, it is determined that the probability of 3 cures, 2 cures, 1 cure, and no cures are .70, .20, .09, and .01 respectively. What is the mean number of cures that can be expected in a group of three?

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Mean and Standard Deviation of a Random Variable

Randy and Dan are businessmen who spend time on the road weekly (Mon-Fri). They always eat dinner out.

Randy spends money for dinner according to the following probability chart.

X $15 $20 $25 $30

P(X) .2 .4 .25 .15

Dan spends money for dinner according to the following probability chart.

X $20 $25 $30 $35

P(X) .35 .25 .3 .1

1. Find the average amount of money Randy spends.

2. Find the average amount of money Dan spends.

3. Find the standard deviation of the amount of money Randy spends.

4. Find the standard deviation of the amount of money Dan spends.

5. Find the average amount of money they spend for dinner together.

6. Find the standard deviation of the total money they spend for dinner together.

7. Find the average difference in money they spend for dinner.

8. Find the standard deviation of the difference they spend for dinner.

9. If Randy eats out 4 times weekly, find the average amount of money he spends on dinner weekly as well as the standard deviation.

10. If Dan eats out 5 times weekly and on Friday treats himself to wine worth $10, what is the average amount that he spends on dinner weekly.

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9. Standard Deviation of a random variable:

Go back to pages 9 - 12 on the mean of a random variable and use that figure to calculate the standard deviation of the random variable.

Standard Deviation Rules

Go back to page 13 and calculate the questions dealing with standard deviation.