# Integrate a Two-Sided Exponential Density A continuous random variable has probability density `f(x) = 0.5 * exp(-abs(x))` for every real `x`. Compute `P(1 <= X <= 3)` and show the integral used. ### Constraints & Assumptions - Endpoint inclusion does not change the probability for a continuous distribution. - The density is already normalized over the real line. ### Clarifying Questions to Ask - Is an exact expression, a decimal approximation, or both desired? - Should normalization of the density also be verified? ```hint Remove the absolute value on the interval Every `x` between 1 and 3 is positive, so the density has one simple exponential form there. ``` ### What a Strong Answer Covers - The correct definite integral and antiderivative. - Correct handling of `abs(x)` on a positive interval. - An exact result and a sensible numerical check. ### Follow-up Questions - What is `P(abs(X) <= a)` for nonnegative `a`? - What are the mean and variance of this distribution?