Probability Probability DistributionDistribution

Binomial Binomial DistributionDistribution

We call a distribution a We call a distribution a binomial distribution binomial distribution if all of if all of

the following are true the following are true

1.1. There are a There are a fixed number of trialsfixed number of trials, n, which are all , n, which are all

independent.independent.

2.2. There must be There must be exactly two mutually exclusive exactly two mutually exclusive

outcomes in a trialoutcomes in a trial, such as True or False, yes or , such as True or False, yes or

no, success or failure.no, success or failure.

3.3. The The probability of successprobability of success is the is the samesame for each for each

trial. trial.

What is binomial distribution?What is binomial distribution?

The binomial distribution is used to obtain the The binomial distribution is used to obtain the

probability of observing probability of observing xx successes in successes in nn trials, with trials, with

the probability of success on a single trial denoted the probability of success on a single trial denoted

by by pp. .

The formula for the binomial probability mass The formula for the binomial probability mass

function is function is

                                                                                                             

where where

                                     

Probability mass functionProbability mass function

PP((SS) = ) = pp. . PP((FF) = ) = qq = 1- = 1-pp. . nn indicates the fixed number of trials. indicates the fixed number of trials. xx indicates the number of successes (any indicates the number of successes (any

whole number [0,whole number [0,nn]). ]). pp indicates the probability of success for any indicates the probability of success for any

one trial. one trial. qq indicates the probability of failure (not indicates the probability of failure (not

success) for any one trial. success) for any one trial. PP((xx) indicate the probability of getting exactly ) indicate the probability of getting exactly

xx successes in successes in nn trials. trials.

Some features of binomial distributionSome features of binomial distribution

Some statistics

Average = Mean = np

Standard Deviation = )1( pnp

The Bernoulli distribution is a special The Bernoulli distribution is a special

case of the binomial distribution, case of the binomial distribution,

where n=1. where n=1.

Bernoulli distributionBernoulli distribution

Suppose that each time you take a free throw shot, you have a 25% chance of making it.  If you take 15 shots, a. what is the probability of making exactly 5 of them.

SolutionSolution

We haveWe have                n  =  15,        r  =  5        p  = .25        q  =  .75n  =  15,        r  =  5        p  = .25        q  =  .75

P(5) =  0.165 P(5) =  0.165 

There is a 16.5 percent chance of making exactly 5 There is a 16.5 percent chance of making exactly 5 shots.shots.

Example

 

b.  What is the probability of making fewer than 3

shots?

 

Solution

The possible outcomes that will make this happen

are 2 shots, 1 shot, and 0 shots.  Since these are

mutually exclusive, we can add these probabilities.

        P(2)+P(1)+P(0)

        =  .156 + .067 + .013  =  0.236

There is a 24 percent chance of sinking fewer than

3 shots.

Shape of binomial distribution for different Shape of binomial distribution for different values of values of nn and and r.r.

•When p is small (<0.5) , the binomial distribution is When p is small (<0.5) , the binomial distribution is skewed to the right.skewed to the right.

•As p increases and approaches to 0.5, the As p increases and approaches to 0.5, the skewness is less noticeable.skewness is less noticeable.

•When p = 0.5, the binomial distribution is When p = 0.5, the binomial distribution is symmetrical.symmetrical.

•When p is larger than 0.5, the distribution is When p is larger than 0.5, the distribution is skewed to the left.skewed to the left.

•When n increases binomial distribution When n increases binomial distribution approaches to symmetrical.approaches to symmetrical.

Problems with binomial distributionProblems with binomial distribution

The requirement that the probability of the outcome The requirement that the probability of the outcome must be fixed overtime is very difficult to meet in must be fixed overtime is very difficult to meet in practice. practice. In many industrial processes, however, it is In many industrial processes, however, it is extremely difficult to guarantee that this is indeed the extremely difficult to guarantee that this is indeed the case.case.

Again, the outcome of one trial may affect in any way Again, the outcome of one trial may affect in any way the outcome of any other trial. the outcome of any other trial. Consider an Consider an interviewing process in which high-potential interviewing process in which high-potential candidates are being screened for top positions. If the candidates are being screened for top positions. If the interviewer has talked to five unacceptable candidates interviewer has talked to five unacceptable candidates in a row, he may not view the sixth with complete in a row, he may not view the sixth with complete partiality. Therefore, independence is violated.partiality. Therefore, independence is violated.

Hypergeometric DistributionHypergeometric Distribution

1.1. An outcome on each trial of an experiment is classified An outcome on each trial of an experiment is classified

into into one of two mutually exclusiveone of two mutually exclusive categories-a categories-a

success or a failure success or a failure

2.2. The random variable is the number of successes in a The random variable is the number of successes in a

fixed number of trialsfixed number of trials

3.3. The trials are The trials are not independentnot independent. .

4.4. We assume that we sample from a finite population We assume that we sample from a finite population

without replacement. So, the without replacement. So, the probability of a success probability of a success

change for each trialchange for each trial. .

P.M.F of the Hypergeometric Distribution

If If XX is the number of is the number of SS’s in a completely random ’s in a completely random sample of size sample of size nn drawn from a population consisting drawn from a population consisting of of MM SS’s and (’s and (NN––MM)) FF’s, then the probability ’s, then the probability distribution is hypergeometric distribution and is distribution is hypergeometric distribution and is given bygiven by : :

n

N

xn

MN

x

M

NMnxhxXP ),,;()(

Mean and VarianceThe mean and variance of a hypergeometric rv The mean and variance of a hypergeometric rv XX

having pmf having pmf hh((x;n,M,Nx;n,M,N) are:) are:

N

M

N

Mn

N

nNXV

N

MnXE 1

1)(,)(

ExampleIn a lot of 20 units out of the production line, 2 units In a lot of 20 units out of the production line, 2 units

are known to be defective. If the inspector picks a are known to be defective. If the inspector picks a

sample of 3 units at random, what is the distribution sample of 3 units at random, what is the distribution

of the number of defectives in the sample?of the number of defectives in the sample?

What is the probability of having 0, 1, 2 and 3 What is the probability of having 0, 1, 2 and 3

defectives in the sample?defectives in the sample?What is the probability that none of the chosen What is the probability that none of the chosen

sample are defective?sample are defective?

Poisson DistributionPoisson Distribution

The Poisson distribution is often used to model the number of occurrences during a given The Poisson distribution is often used to model the number of occurrences during a given

time interval or within a specified region. time interval or within a specified region.

The time interval involved can have a variety of lengths, e.g., a second, minute, hour, day, The time interval involved can have a variety of lengths, e.g., a second, minute, hour, day,

year, and multiples thereof. year, and multiples thereof.

Some random variables that typically obey the Some random variables that typically obey the Poisson probability law Poisson probability law

The number of misprints on a page (or group of pages) of a The number of misprints on a page (or group of pages) of a bookbook

The number of people in a community living to 100 years of The number of people in a community living to 100 years of ageage

The number of wrong telephone numbers that are dialed in a The number of wrong telephone numbers that are dialed in a dayday

The number of customers entering a post office (bank, store) The number of customers entering a post office (bank, store) in a give time periodin a give time period

The number of vacancies occurring during a year in the The number of vacancies occurring during a year in the supreme courtsupreme court

Poisson DistributionPoisson Distribution

If X is defined to be the number of occurrences of an If X is defined to be the number of occurrences of an

event in a given continuous interval and is event in a given continuous interval and is

associated with a Poisson process with parameter associated with a Poisson process with parameter

>0, then X has a Poisson distribution with pdf:>0, then X has a Poisson distribution with pdf:

The mean and variance of a Poisson random The mean and variance of a Poisson random

variable X are:variable X are:

0,1,2,... x where,!

)(

x

exp

x

µ = µ = 22 = =

5 0 5 1 5 2

5

e 5 e 5 e 5P(X< 3)

0! 1! 2!5

e (1 5 ) 0.5732

For example, the number of vehicles crossing a bridge in a rural area might be For example, the number of vehicles crossing a bridge in a rural area might be

modeled as a Poisson process. If the average number of vehicles per hour, modeled as a Poisson process. If the average number of vehicles per hour,

during the hours of 10:00 AM to 3:00 PM is 20 we might be interested in the during the hours of 10:00 AM to 3:00 PM is 20 we might be interested in the

probability that fewer than three vehicles cross on from 12:30 to 12:45 PM. In this probability that fewer than three vehicles cross on from 12:30 to 12:45 PM. In this

case = (20 per hour)(0.25hours) = 5.case = (20 per hour)(0.25hours) = 5.

Poisson DistributionPoisson Distribution