$12\mathrm{.}7.$ In a simple circuit two resistors $R_1$ and $R_2$ are connected in series. Hence the total resistance is given by $R=R_1+R_2.$ Suppose that $R_1$ and $R_2$ are independent random variables each having the pdf

$12\mathrm{.}8.$ Suppose that the resistors in Problem $12\mathrm{.}7$ are connected in parallel. Find the pdf of $R,$ the total resistance of the circuit $($set up only, in integral form$).[$Hint: The relationship between $R,$ and $R_1,$ and $R_2$ is given by $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}.]]=[\frac{1}{R_1}+\frac{1}{R_2}.$