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How much variation should you have in your blood pressure? How likely is that stock price to double by the end of the year? Use continuous probability distributions to find out!

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?

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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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