Discrete Random Variables: PMF, CDF, and Expected Value
Understand discrete random variables, probability mass functions, and expected values for Exam P.
Probability Mass Function and CDF
A discrete random variable X takes countable values. Its probability mass function (PMF) is p(x) = P(X = x), and must satisfy p(x) >= 0 for all x and the sum of all p(x) = 1. The cumulative distribution function (CDF) is F(x) = P(X <= x) = sum of p(t) for all t <= x.
On Exam P, you may be given a PMF in table form or as a formula and asked to compute probabilities. For instance, if X has PMF p(x) = c * x for x = 1, 2, 3, 4, find c. The sum must equal 1: c(1 + 2 + 3 + 4) = 10c = 1, so c = 0.1.
Expected Value and Variance
The expected value E[X] = sum of x * p(x) measures the center of the distribution. The variance Var(X) = E[X^2] - (E[X])^2 measures spread. The standard deviation is the square root of variance. A useful shortcut for Exam P: Var(aX + b) = a^2 * Var(X).
Continuing the example, E[X] = 0.1(1 + 4 + 9 + 16) = 0.1(30) = 3.0. E[X^2] = 0.1(1 + 8 + 27 + 64) = 10, so Var(X) = 10 - 9 = 1.
Common Exam P Patterns
Exam P frequently tests whether you can set up expectations correctly. A key identity is E[g(X)] = sum of g(x) * p(x). This is called the Law of the Unconscious Statistician (LOTUS). For example, if you need E[X(X-1)], compute it directly from the PMF rather than finding the distribution of X(X-1) first. Also remember E[X(X-1)] = E[X^2] - E[X], which connects to factorial moments.