On Main Street, a driver’s speed just before an accident is uniformly distributed on [5, 20]. Given the speed, the resulting loss from the accident is exponentially distributed with mean equal to three times the speed.

Calculate the variance of a loss due to an accident on Main Street.

• 525
• 1463
• 1575
• 1632
• 1744

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

Let S be the speed and X be the loss. Given S, X has an exponential distribution with mean 3X. Then, noting that the variance of an exponential random variable is the square of the mean, the variance of a uniform random variable is the square of the range divided by 12, and for any random variable the second moment is the variance plus the square of the mean:

[[math]] \begin{align*} \operatorname{Var}(X) &= \operatorname{Var}[\operatorname{E}(X | S) ] + \operatorname{E}[ \operatorname{Var}(X | S) ] \\ &= \operatorname{Var}[3S] + \operatorname{E}[9S^2] \\ &= 9(20-5)^2/12 + 9[(20-5)^2/12 + 12.5^2] \\ &= 1743.75 \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.