Random Variables

 Teaching Tips

• You can represent the pdf of a discrete random variable with a table (values versus probabilities), a graph, or a formula (e.g. the binomial formula). All three are instructive.

• In contrast, a continuous pdf has only two representations: a graph and a formula. But the formulas for continuous pdfs (e.g. the normal pdf) are not important for AP students; they will use tables or technology rather than calculus to find areas under a curve.

• The terms "mean of a random variable" and "expected value of a random variable" are interchangeable. But students will mistakenly find this mean using the Descriptive Statistics version of an average (x-bar). Explain to them that the formula for an expected value (of a discrete random variable) is a weighted average, and the probabilities are the weights.

• Probability mass functions, probability density functions, probability distribution functions, pdfs, are all names for the same things. "Probability distribution function" is the most general, and "pdf" is the most vague.

• Technology makes it too easy for students to push a button and find probabilities using the "pdf" or "cdf" options on the calculator. Make sure your students understand the relationships between probabilities and the density curves. We recommend that you have your students always sketch and accurately label pictures of the curves and shade the appropriate regions.

• Students should not ever use the cdf option of the calculator for doing discrete probability calculations, because it makes it too easy for them to confuse P(X < a) with P(X <= a). Instead, they should use discrete pdfs to calculate cdfs. We suggest students calculate these probabilities using tables containing the discrete probabilities. They should create these tables themselves, and the calculator (using the list option) makes this fairly straight-forward. It is still productive to require students to make a sketch of the discrete pdf.