Question

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

Short answer

The probability that it will be working

after an additional$8$year is $e^{-1}$

Step-by-step solution

Given Information.

The number of years a radio function is exponentially distributed with parameters$\lambda =\frac{1}{8}$.

Explanation.

Let $X$be the number of years a radio functions. We do not know the age of Jones's second-hand ratio-but given that it is still functioning, we can assume its age is the expected value $X$given by$E\left(X\right)=\frac{1}{\lambda }=\frac{1}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}=8$

Explanation.

Hence, the conditional probability that Jones's ratio functions for another year would be given by

$P(X\ge 16|X\ge 8)=\frac{P(X\ge 16;X\ge 8)}{P(X\ge 8)}$

Explanation.

Of Course,$P(X\ge 16;\mathrm{X}\ge 8)=\mathrm{P}(\mathrm{X}\ge 16)$

Explanation.

Thus $P(X\ge 16|X\ge 8)=\frac{P(X\ge 16)}{P(X\ge 8)}$

$$\begin{gathered} =\frac{{\mathrm{e}}^{\frac{-1}{8}.16}}{{\mathrm{e}}^{\frac{-1}{8}.8}} \\ ={\mathrm{e}}^{-1} \end{gathered}$$