What Is Probability Distribution? Ir. Muhril A., M.Sc., Ph.D. 1

Chapter 6. Discrete Probability Distributions

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 2

What Is A Probability Distributions Probability distributions gives the entire range

of values that can occur based on an experiment.

Probability distributions:

A listing of all the outcomes of an experiment and the probability associated with each outcome.

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 3

Example:Suppose we are interested in the number of heads showing face up on three tosses of a coin. This is the experiment. The possible results are: - zero heads, - One head, - Two heads, and - Three heads.

What is the probability distribution for the number of heads?

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 4

The Results Are:Coin Toss

Possible Result First Second Third Number Of Heads

1 T T T 0

2 T T H 1

3 T H T 1

4 T H H 2

5 H T T 1

6 H T H 2

7 H H T 2

8 H H H 3

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 5

Probability Distribution For The Event Of Zero, One, Two, And Three Heads From Three Tosses Of Coin

Number Of Heads (X) Probability Of Outcome , P(X)

0 1/8 = 0.125

1 3/8 = 0.375

2 3/8 = 0.375

3 1/8 = 0.125

Total 8/8 = 1.000

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 6

What Is A Probability Distributions (continued) Character Of A Probability Distribution:

1. The probability of a particular outcome

is between 0 and 1 inclusive.

2. The outcome are mutually exclusive.

3. The sum of the probabilities of the

various events is equal to 1.

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 7

Random Variables Random Variable:

a quantity resulting from an experiment that, by chance, can assume different values.

A random variable may be either discrete or continuous.

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 8

Random Variables (continued) Discrete Random Variable: - that can assume only certain clearly separated values.

Continuous Random Variable: - can assume one of an infinitely large number of values, within certain limitations.

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 9

The Mean, Variance, And Standard Deviation Of A Discrete Probability Distributions Mean: - represent the central location of a probability distributions. - the long run average value of the random variable. - referred to as its expected value.

Equation 6-1 Page 185 (Lind). = {X.P(X)}

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 10

The Mean, Variance, And Standard Deviation Of A Discrete Probability Distributions (continued)

Variance And Standard Deviation:

Equation 6-2 Page 185 (Lind).

2 = {(X – )2P(X)}

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 11

Example:John Ragsdale sells new cars for Pelican Ford.

John usually sells the largest number of cars on

Saturday. He has developed the following

probability distribution for the number of cars he

expects to sell on a particular Saturday

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 12

Example:Number Of Cars Sold (X) Probability, P(X)

0 0.10

1 0.20

2 0.30

3 0.30

4 0.10

Total 1.00

What Is Probability Distribution?

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Example: On a typical Saturday, how many cars does

John expect to sell?

= {X.P(X)}

= 0(0.10) + 1(0.20) + 2(0.30) + 3(0.30) +

4(0.10)

= 2.1

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 14

Binomial Probability Distribution Widely occurring discrete probability distribution. There are only two possible outcomes on a particular trial of an

experiment. The outcomes are mutually exclusive.

Equation 6-3 Page 190 (Lind). P(X) = nCx x (1-)n-x

P(X) = binomial probability n = the number of trials C = combination x = the random variable defined as the number of successes = the probability of a success on each trial

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 15

Binomial Probability Distribution (continued) Mean Of A Binomial Distribution:

- equation 6-4 Page 191 (Lind).

= n

Variance Of A Binomial Distribution:

- equation 6-5 Page 191 (Lind).

2 = n(1-)

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 16

Example:There are five flights daily from Pittsburg via US Airways into the Bradford, Pennsylvania, Regional Airport. Suppose the probability that any flight arrives late is 0.20. - What is the probability that none of the flight

are late today? - What is the probability that exactly one of the

flights is late today?

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 17

Example: = 0.20

n = 5

x = the number of successes

case a success is a plane that arrives

late.

What Is Probability Distribution?

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Example No late arrivals (x = 0)

P(0) = 5C0(0.20)0(1-0.20)5-0 = 0.3277

The probability that exactly one of the five flights will arrive late today is:

P(1) = 5C1(0.20)1(1-0.20)5-1 = 0.4096

What Is Probability Distribution?

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Hypergeometric Probability Distribution An outcome on each trial of an experiment is classified into

one of two mutually exclusive categories, a success or a failure.

The random variable is the number of successes in a fixed number of trials.

The trials are not independent.

We assume that we sample from a finite population without replacement and n/N>0.05.

What Is Probability Distribution?

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Hypergeometric Distribution: P(x) = [(sCx) (N-sCn-x)] / (NCn) s = the number of successes in the population C = combination x = the number of successes in the sample N = the size of population n = the size of the sample or the number of trials

What Is Probability Distribution?

Ir. Muhril A., M.Sc., Ph.D. 21

Poisson Probability Distribution Describes the number of times some event occurs

during a specified interval.

The interval may be time, distance, area, or volume.

Equation 6-7 Page 203 (Lind). Equation 6-8 Page 204 (Lind).

Example Page 204.