Q.5.33 - A First Course in Probability

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?

Step-by-Step Solution

Answer

The probability that it will be working

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

Step 1 Given Information.

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

Step 2 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$

Step 3 Explanation.

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

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

Step 4 Explanation.

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

Step 5 Explanation.

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

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