What is a cumulative distribution function (CDF)?

• A. F(x) = P(X ≤ x); nondecreasing, right-continuous, ranges from 0 to 1.
• B. F(x) = P(X ≤ x) for discrete X only.
• C. F(x) describes the probability density.
• D. F(x) is the inverse of the PDF.

Answer: (A)

The central idea is that a cumulative distribution function tells us the probability that the random variable is at most a given value. It is defined as F(x) = P(X ≤ x). This function must be nondecreasing because increasing the threshold x cannot reduce the chance that X falls at or below it. It is right-continuous, meaning it’s continuous when approached from the right, though it can have jumps at points where X has positive probability mass. As x goes to very small values, F(x) approaches 0, and as x goes to very large values, F(x) approaches 1.

This description fits for any random variable, whether its distribution is discrete, continuous, or mixed. For a discrete X, F has jumps at the possible values, with jump sizes equal to P(X equals that value). For a continuous X with a density, F is smooth and its derivative, where it exists, is the probability density function (F′(x) = f(x)). The density and the CDF are related but distinct concepts.

The choice states the defining expression and these general properties, making it the best description. The other ideas mix up special cases or confuse the CDF with the density or its inverse, which isn’t correct.