A continuous random variable may assign positive probability to a single exact value.

A continuous random variable may assign positive probability to a single exact value.

• A. True
• B. False
• C. It can happen for certain degenerate cases
• D. It depends on the distribution

Answer: (B)

For a continuous random variable, probabilities are defined over intervals, not individual points. If you have a density f(x), then P(a ≤ X ≤ b) = the area under f between a and b. When you pick a single point x, the interval collapses to one value with zero width, so that area is zero. Therefore P(X = x) = 0 for every x. A degenerate distribution, which places all probability on a single value, is not a continuous distribution, so it’s not a counterexample to this rule. Thus, a true continuous random variable cannot assign positive probability to an exact value.

For a continuous random variable, probabilities are defined over intervals, not individual points. If you have a density f(x), then P(a ≤ X ≤ b) = the area under f between a and b. When you pick a single point x, the interval collapses to one value with zero width, so that area is zero. Therefore P(X = x) = 0 for every x. A degenerate distribution, which places all probability on a single value, is not a continuous distribution, so it’s not a counterexample to this rule. Thus, a true continuous random variable cannot assign positive probability to an exact value.