Discrete Probability DistributionsBinomial DistributionPoisson Distribution

Hypergeometric Distribution

Binomial Probability Formula

xnxxnxx

n qpxxn

nqpCxP

!)!(

!)(

Binomial Probability Distribution

By listing the possible values of

x with the corresponding

probability of each, we can

construct a Binomial Probability

Distribution.

Constructing a Binomial Distribution

In a survey, a company asked their workers

and retirees to name their expected sources

of retirement income. Seven workers who

participated in the survey were asked whether

they expect to rely on Pension for retirement

income. 36% of the workers responded that

they rely on Pension only. Create a binomial

probability distribution.

Constructing a Binomial Distribution

044.0)64.0()36.0()0( 7007 CP

173.0)64.0()36.0()1( 6117 CP

292.0)64.0()36.0()2( 5227 CP

274.0)64.0()36.0()3( 4337 CP

154.0)64.0()36.0()4( 3447 CP

052.0)64.0()36.0()5( 2557 CP

010.0)64.0()36.0()6( 1667 CP

001.0)64.0()36.0()7( 0777 CP

x P(x)

0 0.044

1 0.173

2 0.292

3 0.274

4 0.154

5 0.052

6 0.010

7 0.001

P(x) = 1

Notice all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.

Population Parameters of a Binomial Distribution

Mean: = np

Variance: 2 = npq

Standard Deviation: = √npq

Example

In Murree, 57% of the days in a year are cloudy. Find the mean, variance, and standard deviation for the number of cloudy days during the month of June.

Mean: = np = 30(0.57) = 17.1

Variance: 2 = npq = 30(0.57)(0.43) =

7.353

Standard Deviation: = √npq = √7.353

≈2.71

Problem 1Four fair coins are tossed simultaneously. Find

the probability function of the random variable

X = Number of Heads and compute the probabilities

of obtaining:

No Heads

Precisely 1 Head

At least 1 Head

Not more than 3 Heads

Problem 2

If the Probability of hitting a

target in a single shot is 10% and

10 shots are fired independently.

What is the probability that the

target will be hit at least once?

Poisson ProcessThe Poisson Process is a counting

that counts the number of occurrences of some specific event through time. Number of customers arriving to a

counter Number of calls received at a

telephone exchange Number of packets entering a queue

Poisson Probability Distribution

The Poisson probability distribution provides a good model for the probability distribution of the number of ‘rare events’ that occur randomly in time, distance, or space.

Assumptions Poisson Probability Distribution The probability of an occurrence of an event

is constant for all subintervals and independent events

There is no known limit on the number on successes during the interval

As the unit gets smaller, the probability that two or more events will occur approaches zero.

µ = 1

µ = 4

µ = 10

Poisson Probability Distribution

1,2,... 0,xfor,!

)(

x

exf

x

• f(x) = The probability of x successes over a given period of time or space, given µ

• µ = The expected number of successes per time or space unit; µ > 0

• e = 2.71828 (the base for natural logarithms)

Problem 5

Let X be the number of cars per

minute passing a certain point of some

road between 8 A.M and 10 A.M on a

Sunday. Assume that X has a Poisson

distribution with mean 5. Find the

probability of observing 3 or fewer

cars during any given minute.

Problem 7

In 1910, E. Rutherford and H. Geiger

showed experimentally that number of

alpha particles emitted per second in a

radioactive process is random variable

X having a Poisson distribution. If X

has mean 0.5. What is the probability

of observing 2 or more particles during

any given second?

Problem 9Suppose that in the production of 50 Ω

resistors, non-defective items are those that

have a resistance between 45 Ω and 55 Ω and

the probability of being defective is 0.2%. The

resistors are sold in a lot of 100, with the

guarantee that all resistors are non-defective.

What is the probability that a given lot will

violate this guarantee?

Problem 11

Let P = 1% be the probability

that a certain type of light bulb will

fail in 24 hours test. Find the

probability that a sign consisting of

100 such bulbs will burn 24 hours

with no bulb failures.

Multinomial DistributionIf a given trial can result in K outcomes E1,E2, …, Ek

with probabilities p1,p2, …,pk, then the Probability Distribution of the random variables X1,X2, …, Xk, representing the number of occurrences for E1,E2, …, Ek

in n independent trials is

1

...,...,,

n) ,p , ,p ,p ; x, , x,x(

1

1

22

11

21k21k21

k

ii

k

ii

xkk

xx

k

p

nx

pppxxx

nf

ExampleAn airport has three runways. The probabilities that

the individual runways are accessed by a randomly arriving commercial jets are as following:

Runway 1: p1 = 2/9

Runway 2: p1 = 1/6

Runway 3: p1 = 11/18

What is the probability that 6 randomly arriving airplanes are distributed in the following fashion?

Runway 1: 2 airplanes

Runway 2: 1 airplanes

Runway 3: 3 airplanes

Sampling With Replacement

trialsn

obabilityN

Mp

defectiveM

itemsallN

N

M

N

M

x

nxf

xnx

)(Pr

1)(

Hypergeometric Probability Distribution

In cases where the sample size is

relatively large compared to the

population, a discrete distribution

called hypergeometric may be useful.

Sampling Without ReplacementHypergeometric Distribution

xn

MN

x

M

n

N

n

N

xn

MN

x

Mxf /*)(

= Different ways of picking n things from N

= Different ways of picking x defective from M

= Different ways of picking n-x nondefective from N-M

Hypergeometric DistributionMean and Variance

)1(

))((2

2

NN

nNMNnMVariance

N

MnMean

Problem 13

Suppose that a test for extra sensory

perception consists of naming (in any

order) 3 cards randomly drawn from a

deck of 13 cards. Find the probability

that by chance alone, the person will

correctly name (a) no cards, (b) 1 Card,

(c) 2 Cards, and (d) 3 cards.

Quiz # 232 Cptr (B) – 5 NOV 2012

If the Probability of hitting a

target in a single shot is 5% and

20 shots are fired independently.

What is the probability that the

target will be hit at least once?