1Slide

Random VariablesDefinition & Example

Definition: A random variable is a quantity resulting from a random experiment that, by chance, can assume different values.

Example: Consider a random experiment in which a coin is tossed three times. Let X be the number of heads. Let Hrepresent the outcome of a head and T the outcome of a tail.

2Slide

The sample space for such an experiment will be: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH.

Thus the possible values of X (number of heads) are X = 0, 1, 2, 3.

This association is shown in the next slide.

Note: In this experiment, there are 8 possible outcomes in the sample space. Since they are all equally likely to occur, each outcome has a probability of 1/8 of occurring.

Example (Continued)

3Slide

TTT

TTH

THT

THH

HTT

HTH

HHT

HHH

0

1

1

2

1

2

2

3

Sample

Space

X

Example (Continued)

4Slide

The outcome of zero heads occurred only once.

The outcome of one head occurred three times.

The outcome of two heads occurred three times.

The outcome of three heads occurred only once.

From the definition of a random variable, X as defined in this experiment, is a random variable.

X values are determined by the outcomes of the experiment.

Example (Continued)

5Slide

Let x = number of TVs sold at the store in one day,

where x can take on 5 values (0, 1, 2, 3, 4)

Example: JSL Appliances

Discrete random variable with a finite number of values

6Slide

Let x = number of customers arriving in one day,

where x can take on the values 0, 1, 2, . . .

Example: JSL Appliances

Discrete random variable with an infinite sequence of values

We can count the customers arriving, but there is nofinite upper limit on the number that might arrive.

7Slide

Probability Distribution: Definition

Definition: A probability distribution is a listing of all the outcomes of an experiment and their associated probabilities.

The probability distribution for the random variable X (number of heads) in tossing a coin three times is shown next.

8Slide

Probability Distribution for Three Tosses of a Coin

9Slide

RANDOM VARIABLE

.10

.20

.30

.40

0 1 2 3 4

Random Variables

Discrete Probability Distributions

10Slide

Discrete Random Variable Examples

Experiment Random

Variable

Possible

Values

Make 100 sales calls # Sales 0, 1, 2, ..., 100

Inspect 70 radios # Defective 0, 1, 2, ..., 70

Answer 33 questions # Correct 0, 1, 2, ..., 33

Count cars at toll

between 11:00 & 1:00

# Cars

arriving0, 1, 2, ...,

11Slide

The probability distribution for a random variabledescribes how probabilities are distributed overthe values of the random variable.

We can describe a discrete probability distributionwith a table, graph, or equation.

Discrete Probability Distributions

12Slide

The probability distribution is defined by aprobability function, denoted by f(x), which providesthe probability for each value of the random variable.

The required conditions for a discrete probabilityfunction are:

Discrete Probability Distributions

f(x) > 0

f(x) = 1

P(X) ≥ 0ΣP(X) = 1

13Slide

a tabular representation of the probabilitydistribution for TV sales was developed.

Using past data on TV sales, …

Number

Units Sold of Days

0 80

1 50

2 40

3 10

4 20

200

x f(x)

0 .40

1 .25

2 .20

3 .05

4 .10

1.00

80/200

Discrete Probability DistributionsExample

14Slide

.10

.20

.30

.40

.50

0 1 2 3 4Values of Random Variable x (TV sales)

Pro

bab

ilit

y

Discrete Probability Distributions

Graphical Representation of Probability Distribution

15Slide

Discrete Probability Distributions

As we said, the probability distribution of a discrete random variable is a table, graph, or formula that gives the probability associated with each possible value that the variable can assume.

Example : Number of Radios Sold at Sound City in a Weekx, Radios p(x), Probability0 p(0) = 0.031 p(1) = 0.202 p(2) = 0.503 p(3) = 0.204 p(4) = 0.055 p(5) = 0.02

16Slide

Expected Value of a Discrete Random Variable

The mean or expected value of a discrete random

variable is:

xAll

X xxp )(

Example: Expected Number of Radios Sold in a Week

x, Radios p(x), Probability x p(x)

0 p(0) = 0.03 0(0.03) = 0.00

1 p(1) = 0.20 1(0.20) = 0.20

2 p(2) = 0.50 2(0.50) = 1.00

3 p(3) = 0.20 3(0.20) = 0.60

4 p(4) = 0.05 4(0.05) = 0.20

5 p(5) = 0.02 5(0.02) = 0.10

1.00 2.10

17Slide

Variance and Standard Deviation

The variance of a discrete random variable is:

xAll

XX xpx )()( 22

2

XX

The standard deviation is the square root of the variance.

18Slide

Example: Variance and Standard Deviation of the Number of

Radios Sold in a Week

x, Radios p(x), Probability (x - X)2 p(x)

0 p(0) = 0.03 (0 – 2.1)2 (0.03) = 0.1323

1 p(1) = 0.20 (1 – 2.1)2 (0.20) = 0.2420

2 p(2) = 0.50 (2 – 2.1)2 (0.50) = 0.0050

3 p(3) = 0.20 (3 – 2.1)2 (0.20) = 0.1620

4 p(4) = 0.05 (4 – 2.1)2 (0.05) = 0.1805

5 p(5) = 0.02 (5 – 2.1)2 (0.02) = 0.1682

1.00 0.8900

89.02 X

Variance

9434.089.0 X

Standard deviation

Variance and Standard Deviation

µx = 2.10

19Slide

Expected Value and Variance (Summary)

The expected value, or mean, of a random variableis a measure of its central location.

The variance summarizes the variability in thevalues of a random variable.

The standard deviation, , is defined as the positivesquare root of the variance.

Var(x) = 2 = (x - )2f(x)

E(x) = = xf(x)