Chapter 7Chapter 7Special Discrete

Distributions

Binomial DistributionBinomial Distribution

• Each trial has two mutually exclusive possible outcomes: success/failure

• Fixed number of trials (n)• Trials are independent• Probability of success (p) is the same for

all trials• Binomial random variable: X = the

number of successes

Are these binomial distributions?Are these binomial distributions?

1) Toss a coin 10 times and count the number of heads

Yes

2) Deal 10 cards from a shuffled deck and count the number of red cards

No, probability of red does not remain the same

3) Doctors at a hospital note whether babies born to mothers with type O blood also have type O blood

No, number of trials isn't fixed

Toss a 3 coins and count the number of headsToss a 3 coins and count the number of heads

Construct the discrete probability distribution.

x 0 1 2 3

P(x) .125 .375 .375 .125

Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

Binomial Formula:Binomial Formula:

P(X k) n

k

pk 1 p n k

Where:

n

k

nCk

Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

P(X 2) 3

2

0.52 0.5 1 .375

The number of inaccurate pistons in a group of four is a binomial random variable. If the probability of a defect is 0.1, what is the probability that only 1 is defective?

More than 1 is defective?

P(X 1) 4

1

0.11 0.9 3 .2916

P(X 1) 1 (P(0) P(1)) .0523

CalculatorCalculator

• binompdf(n, p, x) P(X = x)

• binomcdf(n, p, x) P(X < x)

Cumulative probabilities from P(0) to P(x)

A genetic trait in one family manifests itself in 25% of the offspring. If eight offspring are randomly selected, find the probability that the trait will appear in exactly three of them.

At least five of them?

P(X 3) binompdf (8,.25,3) .2076

)5(1)5( XPXP

0273.)4,25,.8(1)5( binomcdfXP

)4(1)5( XPXP

In a certain county, 30% of the voters are Democrats. If ten voters are selected at random, find the probability that no more than six of them will be Democrats.

P(X < 6) = binomcdf(10, .3 ,6) = .9894

What is the probability that at least 7 are notnot Democrats?

P(X > 7) = 1 – binomcdf(10, .7 ,6) = .6496

Skewed right Symmetrical at p =.5 Skewed left

What happened to the shape of the distribution as the probability of success increased?

What do you notice about the means and standard deviations?

As p increases,

• the means increase

• the standard deviations increase until p = .5, then decrease

Binomial Mean and Standard Binomial Mean and Standard DeviationDeviation

X np

X np 1 p

In a certain county, 30% of the voters are Democrats. How many Democrats would you expect in ten randomly selected voters?

What is the standard deviation for this distribution?

X 10(.3) 3 Democrats

X 10(.3)(.7) 1.45 Democrats

expect

Geometric DistributionGeometric Distribution

• Two mutually exclusive outcomes• Each trial is independent• Probability of success remains constant• Random variable: X = number of trials

UNTIL the FIRST success

So what are the possible values of X?

X 1 2 3 4

How far will this go?

. . .

To infinity

• Geometric: NOT a fixed number of trials no "n"

• Binomial starts with 0; Geometric starts with 1

• Binomial dist.: finite; Geometric dist.: infinite

Differences between Binomial & Differences between Binomial & GeometricGeometric

Count the number of boys in a family of four children.

Binomial:

X 0 1 2 3 4

Count children until first son is born

Geometric:

X 1 2 3 4 . . .. . .

Geometric FormulasGeometric Formulas

P(X x) p 1 p x 1

X 1

p

X 1 pp2

Not on green sheet – they will be given if

needed on a test

Calculator

• P(X = x) = geometpdf(p, x)

• P( X < x) = geometcdf(p, x) Cumulative probability from 1 to x

No “n” because there is no fixed number of trials

What is the probability that the first son is the fourth child born?

What is the probability that the first son is born in at most four children?

P(X 4) geometpdf (.5,4) .0625

P(X 4) geometcdf (.5,4) .9375

A real estate agent shows a house to prospective buyers. The probability that the house will be sold is 35%. What is the probability that the agent will sell the house to the third person she shows it to?

How many prospective buyers does she expect to show the house to before someone buys the house?

P(X 3) geometpdf (.35,3) .1479

X 1

.352.86buyers

Poisson DistributionPoisson Distribution• Deals with infrequent events

Examples:• Accidents per month at an intersection• Tardies per semester for a student• Runs per inning in a baseball game

PropertiesProperties

• A discrete number of events occur in a continuous interval

• Each interval is independent of other intervals

• P(success) in an interval is the same for all intervals of equal size

• P(success) is proportional to the size of the interval

FormulasFormulas

X = # of events per unit of time, space, etc.

λ (lambda) = mean of X

P(X x) xe

x!X

X

The average number of accidents in an office building during a four-week period is 2. What is the probability that there will be one accident in the next four-week period?

What is the probability that there will be more than two accidents in the next four-week period?

3233.)2(1)2( XPXP

2707.)1,2()1( poissonpdfXP

The number of calls to a police department between 8 pm and 8:30 pm on Friday averages 3.5.

• What is the probability of no calls during this period?

• What is the probability of no calls between 8 pm and 9 pm on Friday night?

• What is the mean and standard deviation of the number of calls between 10 pm and midnight on Friday night?

P(X = 0) = poissonpdf(3.5, 0) =.0302

P(X = 0) = poissonpdf(7, 0) =.0009

8:00 to 8:30 is a 30 minute interval.8:00 to 9:00 is a 60 minute interval.

Since the interval is doubled, we double the mean amount of calls to keep it proportional.

μ = 14 & σ = 3.742

Be sure to adjust λ!

Let's examine histograms of the Poisson distribution.

λ = 2 λ = 4

λ = 6

What happens to the shape?

What happens to the mean?

What happens to the standard deviation?

As λ increases,As λ increases,

• Distribution becomes more symmetrical

• Mean and standard deviation both increase