Random Variables for AP Statistics

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By — McGraw-Hill Professional
Updated on Feb 3, 2011

Practice problems for these concepts can be found at:

Recall the definition of an experiment (random phenomenon): An activity whose outcome we can observe and measure, but for which we can't predict the result of any single trial. A random variable, X, is a numerical value assigned to an outcome of a random phenomenon. Particular values of the random variable X are often given small case names, such as x. It is common to see expressions of the form P(X = x), which refers to the probability that the random variable X takes on the particular value x.

example: If we roll a fair die, the random variable X could be the face-up value of the die. The possible values of X are {1, 2, 3, 4, 5, 6}. P(X = 2) = 1/6.
example: The score a college-hopeful student gets on her SAT test can take on values from 200 to 800. These are the possible values of the random variable X, the score a randomly selected student gets on his/her test.

There are two types of random variables: discrete random variables and continuous random variables.

Discrete Random Variables

A discrete random variable (DRV) is a random variable with a countable number of outcomes. Although most discrete random variables have a finite number of outcomes, note that "countable" is not the same as "finite." A discrete random variable can have an infinite number of outcomes. For example, consider f (n) = (0.5)n. Then f(1) = 0.5, f(2) = (0.5)2 = 0.25, f(0.5)3 = 0.125,… There are an infinite number of outcomes, but they are countable in that you can identify f(n) for any n.

example: the number of votes earned by different candidates in an election.
example: the number of successes in 25 trials of an event whose probability of success on any one trial is known to be 0.3.

Continuous Random Variables

A continuous random variable (CRV) is a random variable that assumes values associated with one or more intervals on the number line. The continuous random variable X has an infinite number of outcomes.

example: Consider the uniform distribution y = 3 defined on the interval 1 ≤ x ≤ 5. The area under y = 3 and above the x axis for any interval corresponds to a continuous random variable. For example, if 2 ≤ x ≤ 3, then X = 3. If 2 ≤ x ≤ 4.5, then X = (4.5 – 2)(3) = 7.5. Note that there are an infinite number of possible outcomes for X.

Probability Distribution of a Random Variable

A probability distribution for a random variable is the possible values of the random variable X together with the probabilities corresponding to those values.

A probability distribution for a discrete random variable is a list of the possible values of the DRV together with their respective probabilities.

example: Let X be the number of boys in a three-child family. Assuming that the probability of a boy on any one birth is 0.5, the probability distribution for X is

The probabilities Pi of a DRV satisfy two conditions:

  1. 0 ≤ Pi ≤ 1 (that is, every probability is between 0 and 1).
  2. Pi = 1 (that is, the sum of all probabilities is 1).
  3. (Are these conditions satisfied in the above example?)

The mean of a discrete random variable, also called the expected value, is given by

The variance of a discrete random variable is given by

The standard deviation of a discrete random variable is given by

example: Given that the following is the probability distribution for a DRV, find P(X = 3).

solution: Since ∑Pi = 1, P(3) = 1 – (0.15 + 0.2 + 0.2 + 0.35) = 0.1.
example: For the probability distribution given above, find μx and σx.

example: Redo the previous example using the TI-83/84, or equivalent, calculator.

solution: Enter the x values in a list (say, L1) and the probabilities in another list (say, L2). Then enter "1-Var Stats L1,L2" and press ENTER. The calculator will read the probabilities in L2 as relative frequencies and return 4.5 for the mean and 1.432 for the standard deviation.

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