A ____ describes how probability is distributed across values of X.

A ____ describes how probability is distributed across values of X.

• A. PMF
• B. CDF
• C. MGF
• D. PDF

Answer: (D)

Probability density function describes how probability is distributed across the values of X. For a continuous random variable, probability is spread along the real line, and the density tells you how thick the probability is at each point. The probability that X falls within an interval is the area under the density curve over that interval, and the total area under the curve equals 1; a single point has probability zero in continuous cases. The other functions describe related ideas in different ways: the probability mass function assigns probabilities to discrete X values; the cumulative distribution function gives P(X ≤ x); and the moment generating function summarizes moments but isn’t the distribution itself. So the PDF is the function that directly describes how probability is distributed across X.

Probability density function describes how probability is distributed across the values of X. For a continuous random variable, probability is spread along the real line, and the density tells you how thick the probability is at each point. The probability that X falls within an interval is the area under the density curve over that interval, and the total area under the curve equals 1; a single point has probability zero in continuous cases. The other functions describe related ideas in different ways: the probability mass function assigns probabilities to discrete X values; the cumulative distribution function gives P(X ≤ x); and the moment generating function summarizes moments but isn’t the distribution itself. So the PDF is the function that directly describes how probability is distributed across X.