The number of years a radio functions is exponentially distributed with parameter $\lambda =1/8$. If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?


Let $T$ denote the number of years a radio functions. Given that $T$ is exponentially distributed with $\lambda = 1/8$.

The pdf of $T$ is

$$ \begin{aligned} f(t) &= \frac{1}{8}e^{-t/8},\; t>0. \end{aligned} $$
The distribution function of $T$ is

$$ \begin{aligned} F(t) &= P(T\leq t) = 1- e^{-t/8}. \end{aligned} $$
We do not know how long the radio has already been alive. Let $u$ be the number of years that the radio has been alive.

Here we want to find $P(T>8|T>u)$. Using the memoryless property of an exponential distribution, for some fixed $u$ and $s$,

$$ P(T>s+u|T>u) = P(T>s) $$
Thus for unknown $u$, and $s=8$ we have

$$ \begin{aligned} P(T> 8) &= 1- P(T\leq 8)\\ & = 1- F(8)\\ & = 1- \big[1- e^{-8/8}\big]\\ &= e^{-1}\\ & = 0.3679 \end{aligned} $$

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