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

•Gives the values associated with each possible x value

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

Properties for a Properties for a discrete probability discrete probability

distributiondistribution

1)For every possible x value, 0 < P(x) < 1.

2) 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 function of a Linear function of a random variablerandom variable

If x is a random variable and a and b are numerical constants, then the random variable y is defined by

• andbxay

xyxbxay

xbxay

bb

ba

or2222

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

222

21

21y

21

...

...

then...If

xnxxy

xnxx

nxxxy

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