What is the difference between PMF and PDF?

• A. PMF gives probabilities for discrete values; PDF describes a density for continuous values.
• B. PMF describes a density for continuous values; PDF gives probabilities for discrete values.
• C. PMF and PDF are the same.
• D. PMF integrates to 1; PDF does not.

Answer: (A)

Think of how probability is assigned depending on whether outcomes are countable or not. For a discrete random variable, you can list every possible value and the exact chance of each one. That’s a probability mass function: it gives P(X = k) for each discrete value k, and all those probabilities add up to 1.

For a continuous random variable, you can’t assign a positive probability to a single exact value, because there are infinitely many possible outcomes. Instead you describe how dense the probability is across values with a probability density function. The probability that X falls in an interval is found by integrating the density over that interval: P(a ≤ X ≤ b) = ∫ from a to b of f(x) dx. The density at a single point doesn’t give a probability by itself, since P(X = x) = 0 for continuous distributions. The area under the entire density curve across all possible values equals 1, though the density values can be greater than 1.

So the correct idea is that the PMF assigns probabilities to discrete values, while the PDF describes a density for continuous values. The other statements mix up discrete versus continuous or misstate how probabilities relate to the functions.