A couple takes out a medical insurance policy that reimburses them for days of work missed due to illness. Let [math]X[/math] and [math]Y[/math] denote the number of days missed during a given month by the wife and husband, respectively. The policy pays a monthly benefit of 50 times the maximum of [math]X[/math] and [math]Y[/math], subject to a benefit limit of 100. [math]X[/math] and [math]Y[/math] are independent, each with a discrete uniform distribution on the set {0,1,2,3,4}.

Calculate the expected monthly benefit for missed days of work that is paid to the couple.

• 70
• 90
• 92
• 95
• 140

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

Solution: B

Each (x,y) pair has probability 1/25. There are only three possible benefit amounts:

0: Occurs only for the pair (0.0) and so the probability is 1/25.

50: Occurs for the three pairs (0,1), (1,0), and (1,1) and so the probability is 3/25.

100: Occurs in all remaining cases and so the probability is 21/25.

The expected value is 0(1/25) + 50(3/25) + 100(21/25) = 2250/25 = 90.

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