Chapter 7Chapter 7Discrete

Distributions

Random Variable Random Variable --•A numerical variable whose value depends on the outcome of a chance experiment

Two types:Two types:•Discrete – count of some random variable

•Continuous – measure of some random variable

Discrete Probability Discrete Probability DistributionDistribution

1) Gives the probabilities associated with each possible x value

2) Usually displayed in a table, but can be displayed with a histogram or formula

Discrete probability Discrete probability distributionsdistributions

3)For every possible x value,

0 < P(x) < 1.

4) For all values of x,

P(x) = 1.

Suppose you toss 3 coins & record Suppose you toss 3 coins & record the number of heads.the number of heads.

The random variable X defined as ...

Create a probability distribution.

Create a probability histogram.

The number of heads tossed

X 0 1 2 3

P(X) .125 .375 .375 .125

Let x be the number of courses for which a randomly selected student at a certain university is registered.

X 1 2 3 4 5 6 7 P(X) .02 .03 .09 ? .40 .16 .05

P(x = 4) =

P(x < 4) =

P(x < 4) =

What is the probability that the student is registered for at least five courses?

Why does this not start at zero?

.25

.14

.39 P(x > 5) = .61

Formulas for mean & Formulas for mean & variancevariance

ixix

iix

px

px

22

Found on formula card!

Let x be the number of courses for which a randomly selected student at a certain university is registered.

X 1 2 3 4 5 6 7

P(X) .02 .03 .09 .25 .40 .16 .05

What is the mean and standard deviations of this distribution?

4.66 & = 1.2018

Here’s a game:

If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair?

A fair game is one where the cost to play EQUALS the expected

value!X 0 5 20

P(X) 7/9 1/61/18

NO, since = $1.944 which is less than it cost to play ($3).

Linear transformation of Linear transformation of a random variablea random variable

If x is a random variable, then the random variable y is defined by

and

bxay

xy

xy

b

ba

The mean is changed by addition &

multiplication! The standard deviation is

ONLYONLY changed by

multiplication!

Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed? = $622.40 & = $75.60

Linear combinationsLinear combinations

22

ba

baba

baba

ba

Just add or subtract the means!

If independent, always addadd the variances!

A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be

mean SD

MC 38 6

FR 30 7

If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test?

68 & = 9.2195

Special Discrete

Distributions

Binomial Distribution Binomial Distribution B(n,p)B(n,p)1. Each trial results in one of two

mutually exclusive outcomes. (success/failure)

2. There are a fixed number of trials3. Outcomes of different trials are

independent4. The probability that a trial results

in success is the same for all trialsThe 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

Binomial Formula:Binomial Formula:

knk ppk

nkxP

1)(

Where:

knCk

n

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

375.5.05.02

3)2(

)5,.3(

12

xP

B

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

More than 1 is defective?

2916.9.01.01

4)1(

)1,.4(

31

xP

B

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(

)25,.8(

binompdfXP

B

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

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.

B(10,.3)

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(.10

sRepublican3)3(.10

)3,.10(

x

x

B

expect

In a certain county, 30% of the voters are Republicans. What is the probability that the number of Republicans out of 10 is within 1 standard deviation of the mean?

B(10,.3)

P(1.55 < x < 4.45) = binomcdf(10,.3,4) – binomcdf(10,.3,1) = .7004

Geometric Geometric Distributions:Distributions:

1.There are two mutually exclusive outcomes

2.Each trial is independent of the others

3.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

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

1

1)(

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)

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 born is at most the fourth child?

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

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

What is the probability that the first son born is at least the third child? 25.)2,5(.1)3( 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? SD?

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

buyers86.235.1

x buyers304.235.

35.12

x