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probabilitylect2

S, need not be numbers. For example, in tossing a coin the outcomes are H (heads) or T    (tails), and in tossing

a pair of dice the outcomes are pairs of integers. However, we frequently wish to assign a speci?c number to

each outcome of the experiment. For example, in coin tossing, it may be convenient to assign 1 to H        and 0

to T ; or, in the tossing of a pair of dice, we may want to assign the sum of the two integers to the outcome.

Such an assignment of numerical values is called a random variable. More generally, we have the following

de?nition.

De?nition 7.4:   A random variable X is a rule that assigns a numerical value to each outcome in a sample space S.

We shall let R_Xdenote the set of numbers assigned by a random variable X, and we shall refer to R_Xas the

range space.

Remark:  In more formal terminology, X is a function from S to the real numbers R, and R_Xis the range of X.

Also, for some in?nite sample spaces S, not all functions from S to R are considered to be random variables.

However, the sample spaces here are ?nite, and every real-valued function de?ned on a ?nite sample space is

a random variable.

EXAMPLE 7.13  A pair of fair dice is tossed. (See Example 7.2.) The sample space S consists of the 36 ordered

pairs (a, b) where a and b can be any of the integers from 1 to 6.

Let X assign to each point in S the sum of the numbers; then X is a random variable with range space

R_X= {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Let Y assign to each point the maximum of the two numbers; then Y is a random variable with range space

R_Y= {1, 2, 3, 4, 5, 6}