Question

for any continuous random variable x, p(x = a) equals ____ for all values a.

Explanation:

Step1: Recall Continuous Random Variable Property

For a continuous random variable \( X \), the probability density function \( f(x) \) is used to find probabilities over intervals. The probability that \( X \) takes a specific value \( a \) is the integral of the density function from \( a \) to \( a \), i.e., \( P(X = a)=\int_{a}^{a}f(x)dx \).

Step2: Evaluate the Integral

The integral of a function from a limit to the same limit is always \( 0 \). So \( \int_{a}^{a}f(x)dx = 0 \), which means \( P(X = a)=0 \) for any continuous random variable \( X \) and any value \( a \).

Answer:

\( 0 \)