In the next two examples, integration is used to find probabilities.

Question

If a continuous random variable, \(X\), has probability density function

\[ f(x) = \begin{cases} 1 - \dfrac x 2 & \quad \text{for } 0 \lt x \lt 2 \\[0.2em] 0 & \quad \text{otherwise} \end{cases} \]

what is the probability of getting a value less than 1?

(Solved in full version)

The next example involves a distribution called an exponential distribution; practical applications of this distribution will be described in the next chapter.

Question

If a continuous random variable, \(X\), has probability density function

\[ f(x) = \begin{cases} 4\;e^{-4x} & \quad \text{for } x \ge 0\\[0.2em] 0 & \quad \text{otherwise} \end{cases} \]

what is the probability of getting a value less than 1?

(Solved in full version)