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{\bf Question}

The lifetime $X$ of an electrical component has an exponential
distribution such that $P(X \leq 1000)=0.75$.  What is the
expected lifetime of the component?

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{\bf Answer}

Given that $X \sim \mathrm{exponential} (\beta)$ and $P\{X \leq
1000\}=0.75$ then the problem is to find $\beta$ since
$E(X)=\beta$.

$f(x|\beta)=\ds \frac{1}{\beta}e^{\frac{-x}{\beta}},\ \ \
0<x<\infty$

For any $\ds a>0, P\{x\le a\}=F(a)=\int_0^a
\frac{1}{\beta}e^{-\frac{y}{\beta}}\, dy = 1-e^{-\frac{a}{\beta}}$


$P\{X \leq 1000\}=1-e^{-\frac{1000}{\beta}}=0.75 \Rightarrow
\beta=721.35$

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