Chapter 7Chapter 7Special Discrete

Distributions

Binomial Distribution Binomial Distribution B(n,p)B(n,p)• Each trial results in one of two

mutually exclusive outcomes. (success/failure)

• There are a fixed number of trials• Outcomes of different trials are

independent• The probability that a trial results in

success is the same for all trials• The binomial random variable x is

defined as the number of successes out of the fixed number

Are these binomial Are these binomial distributions?distributions?1) Toss a coin 10 times and count the

number of headsYes

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

No, probability does not remain constant

3) Two parents with genes for O and A blood types and count the number of children with blood type O

No, no fixed number

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

Find the discrete probability distributionX 0 1 2 3P(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:

knk ppkn

kxP

1)(

Where:

knCkn

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

375.5.05.023)2( 12

xP

The number of inaccurate gauges 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?

2916.9.01.014)1( 31

xP

0523.))1()0((1)1( PPxP

CalculatorCalculator• Binomialpdf(n,p,x) – this

calculates the probability of a single binomial P(x = k)

• Binomialcdf(n,p,x) – this calculates the cumulative probabilities from P(0) to P(k) OR P(X < k)

A genetic trait of 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 5?

2076.)3,25,.8()3( fbinomialpdXP

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

In a certain county, 30% of the voters are Republicans. If ten voters are selected at random, find the probability that no more than six of them will be Republicans.P(x < 6) = binomcdf(10,.3,6) = .9894What is the probability that at least 7 are notnot Republicans?P(x > 7) = 1 - binomcdf(10,.7,6) = .6496

Binomial formulas for Binomial formulas for mean and standard mean and standard

deviationdeviation

pnp

np

x

x

1

In a certain county, 30% of the voters are Republicans. How many Republicans would you expect in ten randomly selected voters?What is the standard deviation for this distribution?

sRepublican45.1)7)(.3(.10sRepublican3)3(.10

x

x

expect

•In L1 – seq(x,x,0,10)•In L2 – binompdf(10, .1 ,L1)•Sketch histogram on board

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

increases, the shape changes from being skewed right to symmetrical at p =.5 to skewed left.

•Calculate the mean and Calculate the mean and standard deviations for each of standard deviations for each of the probabilities the probabilities What do you notice?As the probability of success increase,•the means increase.•the standard deviations increase to p = .5, then decrease. Their values are also symmetrical.

Geometric Geometric Distributions:Distributions:

• There are two mutually exclusive outcomes

• Each trial is independent of the others

• The probability of success remains constant for each trial.

• The random variable x is the number of trials UNTIL the FIRST success occurs.

So what are the possible values of

X

X 1 2 3 4

How far will this go?

. . .

To infinity

•The difference between binomial and geometric properties is that there is NOT a fixed number of trials in geometric distributions!

Differences between Differences between binomial & geometric binomial & geometric

distributionsdistributions

•Binomial random variables start with 0 while geometric random variables start with 1

Other differences:Other differences:

•Binomial distributions are finite, while geometric distributions are infinite

Geometric Formulas:Geometric Formulas:

2

1

1

11)(

pp

p

ppxP

x

x

x

Not on formula sheet – they will be given on quiz

or test

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 . . .. . .

What are the values for

these random variables?

Calculator• geometpdf(p,x) – finds the

geometric probability for P(X = k)

• Geometcdf(p,x) – finds the cumulative probability for P( X < k)

• P(X > k) = 1- geometcdf(p,x-1)

No “n” because

there is no fixed

number!

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

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

0625.)4,5(.)4( dfgeometricpXP

9375.)4,5(.)4( dfgeometriccXP

A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person 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?

1479.)3,35(.)3( dfgeometricpxP

buyers86.235.1

x

Poisson DistributionsPoisson DistributionsThis distribution deals with the

probabilities of rare events that occur infrequently in space, time, distance, area, etc.

Examples:• The number of accidents that

occur per month at a given intersection

• The number of tardies per semester for a given student

• The number of runs per inning in a baseball game

Properties:Properties:• The occurrence of a success in

any interval is independent of that in any other interval

• The probability that a success will occur in any interval is the same for all intervals of equal size and is proportional to the size of the interval

• We observe a discrete number of events in a continuous (fixed) interval.

Formulas:Formulas:X = number of rare events per unit of time, space, etc.= mean value of X (Greek letter lambda)

x

x

x

xeXP !)(

The number of accidents in an office building during a four-week period averages 2. What is the probability 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(...)0((1)2( PPXP

2707.!12)1(

21

eXP

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

From 8:00 until 8:30 is a 30 minute period.

From 8:00 until 9:00 is a 60 minute period.

Since the period is doubled, you must double the mean amount of

calls to keep it proportional!

= 14 & = 3.742

Be sure to adjust !

Examine the histograms of the Poisson distributions – = 2 = 4

= 6

What happens to the shape?What happens to the means?What happens to the standard

deviations?

As As increases increases•The distributions become more symmetrical

•The means increase•The standard deviations increase