**Problem Statement:**A random variable $ X $ has an exponential density function given by:\[f(x) = \begin{cases} 0.2e^{-2x} & \text{for } 0 \leq x \\0 & \text{otherwise}\end{cases}\]Find the probability that $ X $ is between 4 and 6 (to three decimal places).

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A First Course in Probability (10th Edition)

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ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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**Problem Statement:**

A random variable $ X $ has an exponential density function given by:

\[
f(x) =
\begin{cases}
0.2e^{-2x} & \text{for } 0 \leq x \\
0 & \text{otherwise}
\end{cases}
\]

Find the probability that $ X $ is between 4 and 6 (to three decimal places).

Expert Solution

Step 1

The probability that X is between 4 and 6 is,

$\begin{array}{rcl} P (4 \leq x < 6) & = & \int_{4}^{6} f (x) d x \\ = & \int_{4}^{6} 0.2 e^{- 0.2 x} d x \\ = & 0.2 \int_{4}^{6} e^{- 0.2 x} d x \\ = & 0.2 {[- \frac{e^{- 0.2 x}}{0.2}]}_{4}^{6} \\ = & - [e^{- 0.2 \times 6} - e^{- 0.2 \times 4}] \\ = & e^{- 0.8} - e^{- 1.2} \\ = & 0.148 \end{array}$

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