Discrete random variables
Objective. Construct distributions and cumulative functions.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
A discrete random variable is one that takes a countable set of possible values, each with an associated probability given by its probability mass function (pmf), p(x) = P(X = x), which must satisfy p(x) ≥ 0 and Σp(x) = 1 over all possible values. The cumulative distribution function (cdf) F(x) = P(X ≤ x) accumulates these probabilities, forming a non-decreasing step function that jumps at each possible value of X. Constructing a distribution means listing every outcome with its probability, then verifying the probabilities sum to exactly 1 before building the cdf from running totals. For a fair coin flipped twice, X counting heads has p(0) = 1/4, p(1) = 1/2, p(2) = 1/4, giving F(1) = 3/4.
Worked example
A discrete random variable X has pmf P(X=1) = 0.2, P(X=2) = 0.5, P(X=3) = 0.3. Find the cumulative distribution value F(2).
- 1Recall F(2) = P(X ≤ 2), summing all probabilities up to and including 2.
- 2Identify the relevant pmf values: P(X=1) = 0.2 and P(X=2) = 0.5.
- 3Add these probabilities together: 0.2 + 0.5.
- 4Simplify the sum to obtain F(2).
Answer: F(2) = 0.7
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Practice
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Start practisingCommon mistake. Students sometimes compute F(x) = P(X ≤ x) as if it meant P(X = x), confusing the cumulative function with the pmf; F accumulates all probability up to and including x, not just at x.
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