Integrating a Two-Sided Exponential Density

Suppose a continuous random variable has density $f(x) = 0.5 \exp(-|x|)$ for every real $x$. Find $P(1 \le X \le 3)$ and write out the integral that gives it.

Constraints and Assumptions

• For a continuous distribution, including or excluding endpoints does not affect the probability.
• The density is already normalized over the entire real line.

Questions to Clarify

• Do you want an exact expression, a decimal approximation, or both?
• Should the normalization of the density be checked as well?

Hint — Drop the absolute value on this interval: Every x between 1 and 3 is positive, so the density has a single simple exponential form there.

What a Strong Response Should Include

• The proper definite integral and its antiderivative.
• Correct treatment of $|x|$ on an interval where x is positive.
• An exact value and a reasonable numerical check.

Follow-Up Questions

• What is $P(|X| \le a)$ for nonnegative $a$?
• What are the distribution's mean and variance?

Overview: For a random variable with density $0.5 \exp(-|x|)$, compute the probability that it lies between 1 and 3.