# ProbabilityRandom Variables

An event may be regarded as function of the outcome of an experiment: based on the outcome, we can say that the event occurred or didn't occur. We will often be interested in specifying richer information about the outcome of an experiment than a simple yes or no. Specifically, we will often want to specify information in the form of a real number.

For example, suppose that you will receive a dollar for each head flipped in our two-fair-flips experiment. Then your payout might be 0 dollars, 1 dollar, or 2 dollars. Because represents a value which is random (that is, dependent on the outcome of a random experiment), it is called a random variable. A random variable which takes values in some finite or countably infinite set (such as , in this case) is called a discrete random variable.

Since a random variable associates a real number to each outcome of the experiment, in mathematical terms a random variable is a function from the sample space to . Using function notation, the dollar-per-head payout random variable satisfies

Note that a random variable , as a function from to , does not have its own uncertainty: for each outcome , the value of is consistently and perfectly well defined. The randomness comes entirely from thinking of as being selected randomly from . For example, the amount of money you'll take home from tomorrow's poker night is a random quantity, but the function which maps each poker game outcome to your haul is fully specified by the rules of poker.

We can combine random variables using any operations or functions we can use to combine numbers. For example, suppose is defined to be the number of heads in the first of two coin flips. In other words, we define

and is defined to be the number of heads in the second flip. Then the random variable maps each to . This random variable is equal to , since for every .

Exercise
Suppose that the random variable represents a fair die roll and is defined to be the remainder when is divided by .

Define a six-element probability space on which and may be defined, and find for every integer value of .

Solution. We set From the sample space, we see that for any integer value we have

Exercise
Consider a sample space and an event . We define the random variable by

The random variable is called the indicator random variable for If is another event, which of the following random variables are necessarily equal?

XEQUATIONX1XEQUATIONX and XEQUATIONX2XEQUATIONX
XEQUATIONX3XEQUATIONX and XEQUATIONX4XEQUATIONX
and XEQUATIONX5XEQUATIONX

Solution.

• Since if and only if and we see that

• Because may be equal to 2 (on the intersection of and ), we cannot have in general.

• We observe that because if and only if

Bruno