In a continuous distribution, what is P(X = x)?

• A. P(X = x) for continuous X is always positive.
• B. P(X = x) for continuous X is zero.
• C. P(X = x) for continuous X is always equal to 1.
• D. P(X = x) for continuous X is undefined.

Answer: (B)

In a continuous distribution, the probability that X takes an exact value x is zero. Probabilities are defined over intervals, not single points, so the chance that X lies exactly at one precise value corresponds to an interval of zero width, which has probability zero. The density function describes how probability is distributed over values, but it’s not itself the probability of a single point. For example, with X uniformly distributed on [0,1], the density is 1 across the interval, yet P(X = 0.5) = 0.