A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 to 10. One policy has a deductible of 1 and the other has a deductible of 2. The family experiences exactly one loss under each policy.

Calculate the probability that the total benefit paid to the family does not exceed 5.

• 0.13
• 0.25
• 0.30
• 0.32
• 0.42

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

Solution: C

Define X and Y to be loss amounts covered by the policies having deductibles of 1 and 2, respectively. The shaded portion of the graph below shows the region over which the total benefit paid to the family does not exceed 5:

We can also infer from the graph that the uniform random variables [math]X[/math] and [math]Y[/math] have joint density function

[[math]] f(x,y) = \frac{1}{100}, \, 0 \lt x \lt 10, \, 0 \lt y \lt 10 [[/math]]

We could integrate f over the shaded region in order to determine the desired probability. However, since X and Y are uniform random variables, it is simpler to determine the portion of the 10 x 10 square that is shaded in the graph above. That is, [math]\operatorname{P} ( \textrm{Total Benefit Paid Does not Exceed 5})[/math]

[[math]] \begin{align*} &= \operatorname{P}(0 \lt X \lt 6, 0 \lt Y \lt 2) + \operatorname{P}(0 \lt X \lt1, 2 \lt Y \lt 7) + \operatorname{P}( 1 \lt X \lt 6, 2 \lt Y \lt 8-X ) \\ &= \frac{(6)(2)}{100} + \frac{(1)(5)}{100} + \frac{(1/2)(5)(5)}{100} \\ &= \frac{12}{100} + \frac{5}{100} + \frac{12.5}{100} \\ &= 0.295. \end{align*} [[/math]]

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