Cumulative Distribution Functions

Probability · Axiom Academy

LESSON Cumulative Distribution Functions Understanding cumulative probabilities for discrete random variables The Cumulative Distribution Function F(x) gives the probability that the random variable X is less than or equal to x. Unlike the PMF which gives exact probabilities P(X = x), the CDF accumulates probabilities up to and including x. It answers "What's the probability X is at most x?" 2. Step Function for Discrete Variables For discrete random variables, the CDF is a step function that jumps at each possible value. At each value x where P(X = x) > 0, the CDF increases by that probability. Between possible values, the CDF stays constant. The function "steps up" because we're accumulating discrete chunks of probability at specific points. 3. Properties and Relationship to PMF • F(x) is non-decreasing (never goes down) • lim(x→-∞) F(x) = 0 and lim(x→∞) F(x) = 1 Relationship to PMF: F(x) = Σ p(k) for all k ≤ x, and conversely p(x) = F(x) - F(x⁻), where x⁻ is the value just before x.

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