The probability that a continuous random variable equals a specific value c is always 0.

The probability that a continuous random variable equals a specific value c is always 0.

• A. True
• B. False
• C. Cannot be determined
• D. Depends on the distribution

Answer: (A)

In continuous distributions, probabilities come from areas under the density curve, not from single points. The probability that X equals a specific value c is the integral of the density over the interval [c, c], which has length zero. That integral is zero, so P(X = c) = 0 for any c. The density value at c, f(c), can be positive, but it does not provide a positive probability for a single point because a point has no width and thus no area. Other choices don’t fit because this property holds for all continuous distributions, not just some; probabilities are assigned to intervals, and a single point always has zero probability regardless of the shape of the density. Therefore the statement is true.

In continuous distributions, probabilities come from areas under the density curve, not from single points. The probability that X equals a specific value c is the integral of the density over the interval [c, c], which has length zero. That integral is zero, so P(X = c) = 0 for any c. The density value at c, f(c), can be positive, but it does not provide a positive probability for a single point because a point has no width and thus no area.

Other choices don’t fit because this property holds for all continuous distributions, not just some; probabilities are assigned to intervals, and a single point always has zero probability regardless of the shape of the density. Therefore the statement is true.