# Solved: The number of years a radio functions is exponentially distributed with parameter

#### ByDr. Raju Chaudhari

Feb 24, 2021

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?

#### Solution

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}