# Homework 3

Due 3/3/23 at 11:59 PM

## Problem 1

Let the random variable X follow an exponential distribution with \lambda=1.5. Find the probability of X being in between 1 and 2? The pdf of an exponential distribution:

f_X(x) = \left\{\begin{array}{cc} \lambda e^{-x\lambda} & x>0\\ 0 & \mathrm{otherwise} \end{array}\right.

## Problem 2

Find the expected value and variance of an exponential distribution.

## Problem 3

Find the variance of a random variable X that follows a beta distribution.

f_X(x) = \left\{\begin{array}{cc} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} & 0 \le X \le 1 \\ 0 & \mathrm{otherwise} \end{array}\right.