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Outline

• Random variables• Probability distributions• Expected value• Variance and standard deviation of a random variable

LESSON 8: RANDOM VARIABLES EXPECTED VALUE AND VARIANCE

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RANDOM VARIABLES

• Random variable:– A variable whose numerical value is determined by the

outcome of a random experiment• Discrete random variable

– A discrete random variable has a countable number of possible values.

– Example• Number of heads in an experiment with 10 coins• If X denotes the number of heads in an experiment

with 10 coins, then X can take a value of 0, 1, 2, …, 10

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RANDOM VARIABLES

– Other examples of discrete random variable: number of defective items in a production batch of 100, number of customers arriving in a bank in every 15 minute, number of calls received in an hour, etc.

• Continuous random variable– A continuous random variable can assume an

uncountable number of values.– Examples

• The time between two customers arriving in a bank, the time required by a teller to serve a customer, etc.

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DISCRETE PROBABILITY DISTRIBUTION

• Discrete probability distribution– A table, formula, or graph that lists all possible events and

probabilities a discrete random variable can assume – An example is shown on the next slide. – Suppose a coin is tossed twice. The events

• HH: both head• 1H1T: One head, one tail• TT: both tail

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DISCRETE PROBABILITY DISTRIBUTION

0

0.25

0.5

0.75

HH 1H1T TT

Event

Pro

bab

ilit

y

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HISTOGRAM OF RANDOM NUMBERS

– Suppose that the following is a histogram of 40 random numbers generated between 0 and 1.

Histogram

0

1

2

3

4

5

6

7

8

0.125 0.25 0.375 0.5 0.625 0.75 0.875 1

Random Numbers

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qu

ency

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RANDOM NUMBER GENERATION

• Most software can generate discrete and continuous random numbers (these random numbers are more precisely called pseudo random numbers) with a wide variety of distributions

• Inputs specified for generation of random numbers:– Distribution– Average– Variance/standard deviation– Minimum number, mode, maximum number, etc.

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RANDOM NUMBER GENERATION

• Next 4 slides– show histograms of random numbers generated and

corresponding input specification.– observe that the actual distribution are similar to but

not exactly the same as the distribution desired, such imperfections are expected

– methods/commands used to generate random numbers will not be discussed in this course

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RANDOM NUMBER GENERATION: EXAMPLE

– A histogram of random numbers: uniform distribution, min = 500 and max = 800

Uniform Distribution

05

10152025

500

520

540

560

580

600

620

640

660

680

700

720

740

760

780

800

Random Numbers

Fre

qu

ency

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– A histogram of random numbers: triangular distribution, min = 3.2, mode = 4.2, and max = 5.2

RANDOM NUMBER GENERATION: EXAMPLE

Triangular Distribution

020

406080

100120

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2

Radom Numbers

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cy

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– A histogram of random numbers: normal distribution, mean = 650 and standard deviation = 100

RANDOM NUMBER GENERATION: EXAMPLE

Normal Distribution

0

5

10

15

20

25

30

350

390

430

470

510

550

590

630

670

710

750

790

830

870

910

950

Random Numbers

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qu

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cy

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– A histogram of random numbers: exponential distribution, mean = 20

RANDOM NUMBER GENERATION: EXAMPLE

Exponential Distribution

0

10

20

30

40

1 6

11 16

21

26

31

36

41

46

51

56

61

66

71

76

Random Numbers

Fre

qu

en

cy

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CONTINUOUS PROBABILITY DSTRIBUTION

• Continuous probability distribution– Similar to discrete probability distribution– Since there are uncountable number of events, all the events

cannot be specified– Probability that a continuous random variable will assume a

particular value is zero!!– However, the probability that the continuous random variable

will assume a value within a certain specified range, is not necessarily zero

– A continuous probability distribution gives probability values for a range of values that the continuous random variable may assume

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z

f(x)

CONTINUOUS PROBABILITY DSTRIBUTION

15

z

f(x)

CONTINUOUS PROBABILITY DSTRIBUTION

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CONTINUOUS PROBABILITY DSTRIBUTION

• The Probability Density Function– The probability that a continuous random variable will fall

within an interval is equal to the area under the density curve over that range

dxxfbXaPb

a

z

f(x)

a b

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CONTINUOUS PROBABILITY DSTRIBUTION

Example 1: Consider the random variable X having the following probability density function:

Plot the probability density curve and find

Otherwise0

202

xx

xf

15.0 XP

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EXPECTED VALUE AND VARIANCE

• It’s important to compute mean (expected value) and variance of probability distribution. For example, – Recall from our discussion on random variables and

random numbers that if we want to generate random numbers, it may be necessary to specify mean and variance (along with the distribution) of the random numbers.

– Suppose that you have to decide whether or not to make an investment that has an uncertain return. You may like to know whether the expected return is more than the investment.

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EXPECTED VALUE

• The expected value is obtained as follows:

• E(X) is the expected value of the random variable X

• Xi is the i-th possible value of the random variable X

• p(Xi) is the probability that the random variable X will assume the value Xi

n

iii XpXXE

1

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EXPECTED VALUE: EXAMPLE

Example 2: Hale’s TV productions is considering producing a pilot for a comedy series for a major television network. While the network may reject the pilot and the series, it may also purchase the program for 1 or 2 years. Hale’s payoffs (profits and losses in $1000s) and probabilities of the events are summarized below:

What should the company do?

x

p (x )

R ejec t 1 y ea r 2 y ea rs

-1 0 0 5 0 1 5 0

0 .2 0 .3 0 .5

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EXPECTED VALUE: EXAMPLE

Example 2:

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LAWS OF EXPECTED VALUE

• The laws of expected value are listed below:

• X and Y are random variables• c, c’ are constants• E(X), E(Y), and E(c) are expected values of X, Y and c respectively.

tindependen are and if ,)( .5

)(

)( .4

'' .3

.2

.1

YXYEXEXYE

YEXEYXE

YEXEYXE

XcEccXcE

XcEcXE

ccE

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LAWS OF EXPECTED VALUE: EXAMPLE

Example 3: If it turns out that each payoff value of Hale’s TV is overestimated by $50,000, what the company should do? Use the answer of Example 2 and an appropriate law of expected value. Which law of expected value applies? Check.

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LAWS OF EXPECTED VALUE: EXAMPLE

Example 4: Tucson Machinery Inc. manufactures Computer Numerical Controlled (CNC) machines. Sales for the CNC machines are expected to be 30, 36, 42, and 33 units in fall, winter, spring and summer respectively. What is the expected annual sales? Which law of expected value do you use to answer this?

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VARIANCE

• The variance and standard deviation are obtained as follows:

is the mean (expected value) of random variable X• E[(X-)2] is the variance of random variable X, expected

value of squared deviations from the mean

• Xi is the i-th possible value of random variable X

• p(Xi) is the probability that random variable X will assume the value Xi

2

2

1

22

deviation, Standard

Variance,

XX

i

n

iiX XpXXE

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VARIANCE: EXAMPLE

Example 5: Let X be a random variable with the following probability distribution:

Compute variance.

x

p (x )

-1 0 5 2 0

0 .2 0 .3 0 .5

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VARIANCE: EXAMPLE

Example 5:

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SHORTCUT FORMULA FOR VARIANCE

• The shortcut formula for variance and deviation are as follows:

= E(X ), the mean (expected value) of random variable X• E(X 2) is the expected value of X 2 and is obtained as

follows:

• Xi is the i-th possible value of random variable X

• p(Xi) is the probability that random variable X will assume the value Xi

2

222

deviation, Standard

ce,for varian formulaShortcut

XX

X XE

n

iii XpXXE

1

22

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SHORTCUT FORMULA FOR VARIANCE EXAMPLE

Example 6: Let X be a random variable with the following probability distribution:

Compute variance using the shortcut formula.

x

p (x )

-1 0 5 2 0

0 .2 0 .3 0 .5

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SHORTCUT FORMULA FOR VARIANCE EXAMPLE

Example 6:

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LAWS OF VARIANCE

• The laws of expected value are listed below:

• X and Y are random variables• c, c’ are constants• V(X), V(Y), and V(c) are variances of X, Y and c respectively.

YVXVYXV

YVXVYXV

XVccXcV

XVXcV

XVccXV

cV

)(

)( .5

' .4

.3

.2

0 .1

2

2

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LAWS OF VARIANCE: EXAMPLE

Example 7: Let X be a random variable with the following probability distribution:

Compute

Which law of variance applies? Check.

52 XV

x

p (x )

-1 0 5 2 0

0 .2 0 .3 0 .5

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LAWS OF VARIANCE: EXAMPLE

Example 7:

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LAWS OF EXPECTED VALUE: EXAMPLE

Example 8: Let X be a random variable with the following probability distribution:

Compute

Which laws of expected value and variance apply? Check.

52 2 XE

x

p (x )

-1 0 5 2 0

0 .2 0 .3 0 .5

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LAWS OF EXPECTED VALUE: EXAMPLE

Example 8:

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EXPECTED VALUE AND VARIANCE OF A CONTINUOUS VARIABLE

Expected value of a continuous variable:

Variance of a continuous variable:

dxxxfXE

22Var XEdxxfxX

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READING AND EXERCISES

Lesson 8

Reading:

Section 7-1, 7-2, pp. 191-202

Exercises:

7-4, 7-6, 7-15