What is P(X = c) if X is a continuous random variable?

What is P(X = c) if X is a continuous random variable?

• A. P(X=c) is undefined
• B. P(X=c) can be one
• C. P(X=c) is always zero
• D. P(X=c) equals the density at c

Answer: (C)

For a continuous random variable, probabilities are assigned to intervals, not exact points. The probability density function describes density, not direct probabilities. The probability that X equals a single value c is the integral of the density over the single point {c}, which has zero width: P(X = c) = ∫_c^c f(x) dx = 0. The density value f(c) is the height of the curve, used to compute probabilities over intervals via P(a ≤ X ≤ b) = ∫_a^b f(x) dx. As an interval shrinks to a point, the probability approaches zero, since it’s effectively density times an infinitesimal length. The only exception would be a degenerate distribution that puts all mass at c, but that is not a continuous distribution. Therefore, P(X = c) is always zero for a continuous random variable.

For a continuous random variable, probabilities are assigned to intervals, not exact points. The probability density function describes density, not direct probabilities. The probability that X equals a single value c is the integral of the density over the single point {c}, which has zero width: P(X = c) = ∫_c^c f(x) dx = 0. The density value f(c) is the height of the curve, used to compute probabilities over intervals via P(a ≤ X ≤ b) = ∫_a^b f(x) dx. As an interval shrinks to a point, the probability approaches zero, since it’s effectively density times an infinitesimal length. The only exception would be a degenerate distribution that puts all mass at c, but that is not a continuous distribution. Therefore, P(X = c) is always zero for a continuous random variable.