Practice problems for these concepts can be found at:
 Probability and Random Variables Multiple Choice Practice Problems for AP Statistics
 Probability and Random Variable Free Response Practice Problems for AP Statistics
 Probability and Random Variables Cumulative Review Problems for AP Statistics
 Probability and Random Variables Rapid Review for AP Statistics
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 faceup 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 collegehopeful 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 threechild family. Assuming that the probability of a boy on any one birth is 0.5, the probability distribution for X is
The probabilities P_{i} of a DRV satisfy two conditions:
 0 ≤ P_{i} ≤ 1 (that is, every probability is between 0 and 1).
 ∑P_{i} = 1 (that is, the sum of all probabilities is 1).
(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 ∑P_{i} = 1, P(3) = 1 – (0.15 + 0.2 + 0.2 + 0.35) = 0.1.
example: For the probability distribution given above, find μ_{x} and σ_{x}.
solution:
example: Redo the previous example using the TI83/84, or equivalent, calculator.
solution: Enter the x values in a list (say, L1) and the probabilities in another list (say, L2). Then enter "1Var 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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