Chapter 7Discrete

Distributions

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

Two types:• Discrete – count of some random variable

• Continuous – measure of some random variable

• A random variable is discrete if its set of possible values is a collection of isolated points on the number line.

Discrete and Continuous Random Variables

• A random variable is continuous if its set of possible values includes an entire interval on the number line.

We will use lowercase letters, such as x and y, to represent random variables.

Possible values of a discrete random variable

Possible values of a continuous random variable

Examples1. Experiment: A fair die is rolled

Random Variable: The number on the up faceType: Discrete

2. Experiment: A coin is tossed until the 1st head turns upRandom Variable: The number of the toss that the 1st head turns upType: Discrete

3. Experiment: Choose and inspect a number of parts Random Variable: The number of defective partsType: Discrete

4. Experiment: Measure the voltage in a outlet in your roomRandom Variable: The voltageType: Continuous

5. Experiment: Observe the amount of time it takes a bank teller to serve a customerRandom Variable: The timeType: Continuous

Discrete Probability Distribution

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 distributions

3)For every possible x value,

0 < P(x) < 1.

4) For all values of x,

S P(x) = 1.

Suppose you toss 3 coins & record the number of heads. What is the sample space?

The Random Variable X is defined as …

The number of heads tossed

# of heads

X=0 TTT

X=1 HTT,THT,TTH

X=2 HHT,HTH,THH

X=3 HHH

Create a probability distribution.

Create a probability histogram.

X 0 1 2 3

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

Create a probability distribution.X 0 1 2 3P(X) .125 .375 .375 .125

Now we can use the probability distribution table to answer questions about the variable X.

What is the probability of getting exactly 2 heads?

What is the probability of getting at least 2 heads?

What is the probability of getting at least one head?

P(X=2) =.375

P(X>2) = P(X=2) + P(X=3) = .5

P(X>1) = P(X=1) + P(X=2) + P(X=3) = .875

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

Mean and Variance of

Discrete Random Variables

Probability Distributions are also described by measures of central

tendency and variability.

The MEAN of a discrete random variable X is the average of the possible outcomes of X WITH the weights (probabilities). Other names for the MEAN are the WEIGHTED

AVERAGE or the EXPECTED VALUE.

X i ix p

Probability Distributions are also described by measures of central

tendency and variability.

The VARIANCE is an average of the squared deviation of the values of the variable X

from its mean. The STANDARD DEVIATION is the square root of the variance.

∑ (𝒙 𝒊−𝝁𝒙 )𝟐𝒑𝒊

Formulas for mean & variance

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?

=m 4.66 & s = 1.2018

Find the mean and standard deviation for the number of heads out of 3 tosses.

X 0 1 2 3

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

=m 1.5 & s = .866

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!

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?

X 0 5 20

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

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

What is the random variable X?

Make a probability distribution table.

Let’s play a game. A player pays $5 and draws a card from a deck. If he draws the ace of hearts, he is paid $100. For any other ace, he is paid $10, and for any other heart, he is paid $5. If he draws anything else, he gets nothing. Would you be willing to play?What is the random variable X? _____________________

What are the values for the random variable? ____________

Outcome Payout (X) Probability E(X) Deviation

         

         

         

         

An insurance company offers a “death and disability” policy that pays $10,000

when you die or $5000 if you are permanently disabled. It charges a premium of on $50 a year for this benefit. Is the

company likely to make a profit selling such a plan?

PolicyholderOutcome

Payout X

ProbabilityP(x)

Death $10,000

Disability $5000

Neither $0

1000

1

1000

2

1000

997

Suppose that the death rate in any year is 1 out of every 1000 people, and that another 2 out of 1000 suffer some kind of disability.

PolicyholderOutcome

Payout X

ProbabilityP(x)

Death $10,000

Disability $5000

Neither $0

1000

1

1000

2

1000

2

1000

997

1000

997

1000

1E (X) = Expected value = μ = = $10,000( ) + $5000( ) + $0( ) = $10 + $10 + $0

= $20

all possible

values of x

X x p(x)

PolicyholderOutcome

Payout X

ProbabilityP(x)

Death $10,000

Disability $5000

Neither $0

Deviation(x – μ)

(10,000 – 20) = 9980

(5000 – 20) = 4980

(0 – 20) = - 20

1000

2

1000

997

Var(x) = 99802( ) + 49802( ) + (-20)2 ( ) = 149,600

1000

1

1000

1

1000

2

1000

997

Standard deviation (X) = √149,600 = $386.78

Linear function of a random 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 ONLY 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?m = $622.40 & s = $75.60

Linear combinations

222

21

21y

21

...

...

then...If

xnxxy

xnxx

nxxxy

Just add or subtract the means!

If independent, always add the variances!

The mean of the sum of two random variables is the sum of the means.

E(X + Y) = E(X) + E(Y)

The mean of the difference of two random variables is the difference of the means.

E(X - Y) = E(X) - E(Y)If the random variable are independent, the variance of their sums OR difference is always the sum of the variances.

Var(X + Y) = Var(X) + Var(Y)

Mean SD

X 10 2

Y 20 5

What is the mean and standard deviation of:

a) 3X Mean = 30; standard deviation = 6

b) Y + 6 Mean = 26; standard deviation = 5

c) X + Y Mean = 30; standard deviation = = 5.39

d) X - Y Mean = -10; standard deviation = = 5.39

e) X1 + X1 Mean = 20; standard deviation = = 2.83

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?

=m 68 & s = 9.2195

ExampleSuppose x is the number of sales staff needed on a given day. If the cost of doing business on a day involves fixed costs of $255 and the cost per sales person per day is $110, find the mean cost (the mean of x or mx) of doing business on a given day where the distribution of x is given below.

x p(x)1 0.32 0.43 0.24 0.1

Example continued

x p(x) xp(x)1 0.3 0.32 0.4 0.83 0.2 0.64 0.1 0.4

2.1

x2.1

We need to find the mean of y = 255 + 110x

xy 255 110 x 255 110

255 110(2.1) $486

Example continued

2

x

x

0.89

0.89 0.9434

x p(x) (x-)2p(x)1 0.3 0.36302 0.4 0.00403 0.2 0.16204 0.1 0.3610

0.8900

X255 110

2 2 2 2

x(110) (110) (0.89) 10769

We need to find the variance and standard deviation of y = 255 + 110x

X255 110 x110 110(0.9434) 103.77