An insurance company will cover losses incurred from tornadoes in a single calendar year. However, the insurer will only cover losses for a maximum of three separate tornadoes during this timeframe. Let [math]X[/math] be the number of tornadoes that result in at least 50 million in losses, and let [math]Y[/math] be the total number of tornadoes. The joint probability function for [math]X[/math] and [math]Y[/math] is

[[math]] p(x,y) = \begin{cases} c(x + 2y), \, \textrm{for} \, x = 0,1,2,3, \, y = 0,1,2,3, \, x \leq y \\ 0, \, \textrm{otherwise.} \end{cases} [[/math]]

where [math]c[/math] is a constant.

Calculate the expected number of tornadoes that result in fewer than 50 million in losses.

• 0.19
• 0.28
• 0.76
• 1.00
• 1.10

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

Solution: E

The possible events are (0,0), (0,1), (0,2), (0,3), (1,1), (1,2), (1,3), (2,2), (2,3), and (3,3). The probabilities (without c) sum to

0 + 2 + 4 + 6 + 3 + 5 + 7 + 6 + 8 + 9 = 50. 

Therefor c = 1/50. The number of tornadoes with fewer than 50 million in losses is Y – X. The expected value is

(1/50)[0(0) + 1(2) + 2(4) + 3(6) + 0(3) + 1(5) + 2(7) + 0(6) + 1(8) + 0(9)]=55/50 = 1.1.

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