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(1)

Unit 4 - Probability

Distributions

Random Variables & Probability Distributions Binomial Distribution

(2)

RANDOM VARIABLES AND

RANDOM VARIABLES AND

PROBABILITY

PROBABILITY

DISTRIBUTIONS

DISTRIBUTIONS

Random Variables

Probability Distributions

(3)

Random Variables - Definition

A

random variable

random variable

is a variable whose value is

determined by the outcome of a random experiment.

Typically we use capital letters to represent random

variables, and lower case letters to represent

particular values they can take

Eg : The random variable X represents the face of an unbiased

(4)

Random Variables

• There are two types of random variables

Discrete Random VariablesDiscrete Random Variables – can assume only

countable values • Examples

• Number of chairs

• Number of lotto winners

Continuous Random VariablesContinuous Random Variables – can assume any value

in an interval • Examples

• weight

• height

(5)

Probability Distributions

• A probability distributionprobability distribution is:

• A mathematical function that assigns probabilities to the

values of a random variable • Eg:

• A listing of all the outcomes of an experiment and the

probability of each outcome • Eg:

• A function which describes how probabilities are

distributed over the values of a random variable

• Eg: We roll two dice. Let the random variable X = “sum of the

two faces” 5 ..., , 2 , 1 for 25 2 )

(xxxf

(6)

Properties of Probability

Distributions

Each probability MUST be greater than or equal

to zero

f(x) ≥ 0 or P(X = x) ≥ 0

• ie. Probabilities cannot be negative

The sum of ALL probabilities MUST be equal to 1

(7)

Discrete vs. Continuous Probability

Distributions

• The discrete probability distribution lists all

possible values than the variable can assume and corresponding probabilities.

• For a continuous random variable, it is

(8)

Distributions

Example 1Example 1:

 If we roll two dice and let the random variable X

be the sum of the values facing up

a. Calculate the probability distribution of X b. Determine the probability of X = 5

c. Determine the probability of the sum of the

(9)

Example – Discrete Probability

Distributions

Example 2Example 2:

•Let

•be the probability distribution of X.

a. Calculate the probability of X = 3 b. Calculate P(2 ≤ X < 5)

5 ..., , 2 , 1 for 25 2 )

(10)

Summarising Probability Distributions

• Probability distributions of random variables are

summarized in various ways, using the key ideas of centre, spread, and shape.

• The most important measures are:

• Centre — the mean or expected value (”expectation”) • Spread — the variance and the standard deviation

• Shape — various mathematical models, notably the

(11)

Expected Value

The

expected value

expected value

of a random variable is

simply the mean of a random variable

It is denoted as E(X) – read ‘the expectation of

X’

Calculating the expected value:

• Formula: Σ x P(X) or Σ x f(x)

(12)

Variance

• The variancevariance of a random variable measures

the spread of the distribution

• It is denoted by V(X) or Var(X) • Calculating V(X):

• Formula: Σ x2 P(X) – [Σ x P(X)] 2

Translation:

multiply the square of each x value by its

corresponding probability and sum the results

(13)

Example – Expected Value & Variance

Example 1:

Let x be the number of magazines a person reads every

week. Based on a sample survey of adults, the following probability distribution table was prepared:

Calculate the mean and standard deviation.

x 0 1 2 3 4 5

(14)

Example – Expected Value & Variance

Example 2:

• Loraine Corporation is planning to market a new

makeup product. According to the analysis made by the financial department of the company, it will earn an annual profit of $4.5 million if this product has high sales, an annual profit of $1.2 million if the sales are mediocre and it will lose $2.3 million if the sales are low. The probabilities of these

three scenarios are 0.32, 0.51 and 0.17

• Calculate the mean and standard deviation of the

(15)

BINOMIAL DISTRIBUTION

BINOMIAL DISTRIBUTION

(discrete probability distribution)

(discrete probability distribution)

What is the Binomial Distribution (BD)?

(16)

The Binomial Distribution

• One of the most widely used Discrete Probability

Distributions

• Applied to find the probability that an event occurs

x times in n performances of an experiment.

• To apply the Binomial distribution the random

(17)

The Binomial Distribution

(18)

Binomial Distribution – Example I

Individuals with a certain gene have a 0.70

probability of eventually contracting a certain

disease. What is the probability that 5 out of

10 randomly selected persons with the certain

will have the disease?

This is a Binomial Distribution. Why?

• 10 identical trials (testing for the disease)

• Each trial has only 2 possible outcomes (have the disease,

does not have the disease)

(19)

Binomial Distribution – Example II

Suppose a survey in a particular country

indicated that 9 out of 10 cars carry liability

insurance. If four cars in that country are

involved in accidents what is the probability

that …

This is a Binomial Distribution. Why?

• 4 identical trials (4 cars in accidents)

• Each trial has only 2 possible outcomes (have liability

insurance, does not have liability insurance)

• We assume a constant probability (9 out of 10 – 9/10 or

0.9)

(20)

Binomial Distribution

The formula used to calculate the probability

for the binomial distribution is:

Where :

x = probability of interest • p = probability of success • n = number of trials

q = 1 – p (probability of failure) • n-x = number of failures

nCx = number of ways to get x successes in n trials

)

(

X

x

n

C

x

p

x

q

n x

(21)

x

This is calculated

using:

x

n

C

!

)!

(

!

x

x

n

n

C

x n

n denotes the total number of elements

x denotes the number of elements selected per selection

= the number of combinations of n

(22)

Binomial Distribution

NB FormulaNB Formula:

Step 1Step 1

: Calculate

n

C

x

Step 2Step 2

: Calculate

p

x

Step 3Step 3

: Calculate

q

n-x

Step 4Step 4

: Multiply the results from steps 1, 2 and 3

)

(

X

x

n

C

x

p

x

q

n x

(23)

Example I - Binomial Distribution

• Suppose that the probability of breaking your

leg at a ski lodge is 0.20

1. What is the probability that exactly 2 out of 5 people

will break their leg?

2. What is the probability that more than 2 people out of 5

(24)

Example II - Binomial Distribution

1. What is the probability of obtaining exactly 4

heads in 6 flips of a fair coin?

2. What is the probability of obtaining 4 or more

(25)

Example III - Binomial Distribution

Express House Delivery Service guarantees a refund of

all charges if delivery does not arrive by the specified time. It is known that 2% of packages do arrive by the

specified time. Suppose a corporation mails 10 packages on a certain day.

A) Find the probability that exactly one of these ten

packages will not arrive by the specified time.

B) Find the probability that at most one of these ten

(26)

Distribution

• If the random variable X is binomially distributed

then:

• E(X) = np

• Var(X) = npq or np(1-p)

(27)

Example – Binomial Expectation &

Variance

• Suppose experience has shown that 30% of the

television sets sold on hire purchase at a certain store are eventually re-possessed. If 10 TV sets are sold during a week, determine:

(28)

NORMAL DISTRIBUTION

NORMAL DISTRIBUTION

(continuous probability distribution)

(continuous probability distribution)

What is the Normal Distribution (ND)?

The Standard Normal Distribution & Its Probabilities How to Calculate General Probabilities for the ND The Normal Approximation of the Binomial

Distribution

(29)

The Normal Distribution

• The normal distribution is a continuous

random variable

• It is symmetric and bell-shaped

• The mean = median = mode

• Shape of curve depends on population

(30)

The Normal Distribution

• Center of distribution is 

- the mean

• Spread is determined by

 - the standard deviation

• 50% of the scores lie

above the mean and 50% lie below the mean

• Total area under the

(31)

General Normal Distributions

The mean and standard deviation affect the

shape of the normal distribution

Standard Normal

Smaller Standard Deviation

(32)

The Standard Normal Distribution

This is a special case of the normal distribution

Mean =

= 0

Standard Deviation =

=1

(33)

Distribution

• The units for the standard normal distribution curve

are denoted by z and are called z values

• The z-value gives the distance between the mean and

the point represented by z in terms of the standard deviation

• the z-values on the right side of the mean are positive and

(34)

The Standard Normal Tables

The tables give you

the area to the left

of any positive value

The area under the

curve is equal to 1

(35)

Normal Tables

(36)

Normal Tables cont’d

(37)

Normal Tables cont’d

(38)

Steps to Calculate Normal

Probabilities

Step 1

Step 1

: Sketch the normal distribution and indicate

the mean (the middle) of the random variable X

Step 2

Step 2

: Shade the area you want to find

Step 3

Step 3

: Find the corresponding area to the right of

the mean (if needed)

Step 4

Step 4

: Read the value off the table and convert if

(39)

Example I – Normal Distribution

1. What is the probability of X being less than 1.5

2. What is the probability of X being greater than

1.5?

3. What is the probability of X being less than

-0.67?

4. What is the probability of X being greater than

(40)

Example II – Normal Distribution

5. What is the proportion that falls between 0.5 and 1.5?

6. What is the proportion that falls between -0.5 and 1.5?

(41)

Standardization

• To calculate probabilities from a normal

distribution with  0 and  1 (ie. not standard

normal) we have to standardize the distribution

• Once we know the mean and standard deviation

this can be done

• The formula is:

• After the standardization process is complete (ie.

Using the formula above), calculations are conducted as before

(42)

Example - Standardisation

Assume that the length of time, X, between

charges of a cellular phone is normally

distributed with a

• mean of 10 hours and a

• standard deviation of 1.5 hours

Find the probability that the cellular phone will

last for :

(43)

Calculating % aspects

Sometimes the probability (area under the curve) is given

for us to determine the value

Step 1Step 1: Draw the diagram and shade the relevant area

Step 2Step 2: Determine the standardised value

Step 3Step 3: Solve the function for the un-standardised x value

(44)
(45)

Example – Normal Distribution % I

To qualify for the police academy, candidates must

score in the top 10% on a general ability test. The

test has a mean score of 200 and a standard

(46)

• It is known that the life of a calculator

manufactured by Texas Instruments has a

normal distribution with a mean of 54 months and a standard deviation of 8 months. What should the warranty period be to replace a

(47)

using the Normal Distribution I

• When n is large (n ≥ 20), a normal probability

distribution may be used to provide a good

approximation to the probability distribution of a binomial random variable

(48)

using the Normal Distribution II

The Continuity Correction Factor (CCF)The Continuity Correction Factor (CCF)

 This is the value which is added when a discrete distribution (eg. The binomial distribution) is being

approximated by the normal distribution (a continuous distribution)

Ie. we are correcting the discrete distribution so that it can be approximated by the continuous one

The CCF value used is 0.5

Subtract 0.5 from lower limit and add 0.5 to upper

(49)

Approximate the Binomial Distribution I

Step 1Step 1: Calculate the mean and variance of the binomial

distribution in question

Step 2Step 2: Express the binomial probability to be approximated by

(50)

Approximate the Binomial Distribution II

Step 3: Calculate the probability to be

Step 3

approximated using the formula:

Step 4: Sketch the approximating normal

Step 4

distribution and shade the area corresponding

to the probability of the event of interest

Step 5: Find the probability using the ‘Steps

Step 5

to Calculate Normal Probabilities’ slide

npq

np

X

z

The mean of the binomial distribution

(51)

to the Binomial Distribution

In a recent survey conducted for

Money

magazine, 80% of the women surveyed said

that they are more knowledgeable about

investing now than they were five years ago

(

Money

, June 2002). Suppose this result is

true for the current population of all women.

What is the probability that in a random

sample of 100 women, 72 – 76 will say that

(52)

to the Binomial Distribution

According to the 2001 Youth Risk Behaviour

Surveillance by the Centres for Disease Control and

Prevention, 39% of the 10

th

-graders surveyed said

that they watch three or more hours of television on a

typical school day. Assume that this percentage is

true for the current population of all 10

th

-graders.

What is the probability that 86 or more of the 10

th

(53)

Binomial vs. Normal Distributions

Binomial Normal

Variable Type Discrete (assumes only discrete

values)

Continuous (assumes all values within a given interval)

Mean np μ

Variance npq σ2

Symmetry

 Only symmetrical when p = 0.5

 If p = 0.5 the distribution is

skewed but tends to symmetry as n increases

References

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