Probability
# Continuous Probability Distributions

The probability density function for an Exponential distribution with parameter $\lambda > 0$ is given by

$f(x) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0, \\ 0 & x < 0 \end{cases}.$

At what value of $x$ is the probability density function maximized?

What is the expected value of a random variable with probability density function

$f(x) = \begin{cases} 2e^{-2x} & x \ge 0 \\ 0 & x < 0 \end{cases}?$