Unit 7: Random Variables and their PDFs
|Main Concepts | Demonstration | Teaching Tips | Data Analysis & Activity | Practice Questions | Connections | Fathom Tutorial | Milestone|
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. It might be useful to show students a formula to illustrate how this representation of a continuous pdf connects to that of a discrete pdf, but the formulas for continuous pdfs (e.g. the normal pdf) are generally not important for 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.
• Similarly, remind students that the formula for variance of a discrete random variable is also just a weighted average (of the squared distances from the expected value), where 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.
• To further illustrate the difference between modeling relative frequencies of values found within populations and modeling the probabilities associated with values in a random experiment using pdfs, it might be useful to give your students examples using the normal model. For example, you could draw three curves – one representing height, one representing the height of a randomly selected person, and one representing the mean height of a random sample of people – and emphasize how to properly title and label the graph (and axes) and how to interpret the graph (e.g. the meaning of a shaded area).
Student Misconceptions and Confusions• Random variables are always numerical. Some students may incorrectly give "the color of a card dealt from a deck" as an example of a random variable. But a random variable would be "the number of red cards dealt from a deck of cards."
• You might hear that P(X =x) = 0 for a continuous distribution. This means that, when the possible values are over an infinite spectrum, the probability of any one particular value occuring is 0. Remember that for a continuous distribution, the probability is given by the area under the curve, and the area under a single point is 0.