Discrete Mathematics
# 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

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

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