For a continuous variable, the probability that X lies in an interval [a, b] is given by which of the following?

For a continuous variable, the probability that X lies in an interval [a, b] is given by which of the following?

• A. The integral of f(x) from a to b
• B. The value of f(b) minus f(a)
• C. The area under the entire curve
• D. The height of the density at the midpoint

Answer: (A)

For a continuous random variable, probabilities come from areas under its density curve. The probability that X falls in the interval [a, b] is the area under the density f(x) between a and b, which is written as the integral ∫ from a to b of f(x) dx. This integral accumulates the probability mass across the interval and always lies between 0 and 1 because f is nonnegative and the total area under the curve over the entire real line equals 1. Why the other options don’t fit: taking f(b) minus f(a) gives a difference of density values at endpoints, not a probability—density values aren’t probabilities themselves. The area under the entire curve would be 1, not the probability of just [a, b]. The height of the density at the midpoint is a density value, again not a probability mass.

For a continuous random variable, probabilities come from areas under its density curve. The probability that X falls in the interval [a, b] is the area under the density f(x) between a and b, which is written as the integral ∫ from a to b of f(x) dx. This integral accumulates the probability mass across the interval and always lies between 0 and 1 because f is nonnegative and the total area under the curve over the entire real line equals 1.

Why the other options don’t fit: taking f(b) minus f(a) gives a difference of density values at endpoints, not a probability—density values aren’t probabilities themselves. The area under the entire curve would be 1, not the probability of just [a, b]. The height of the density at the midpoint is a density value, again not a probability mass.