The repair costs for boats in a marina have the following characteristics:

Boat type	Number of boats	Probability that repair is needed	Mean of repair cost given a repair	Variance of repair cost given a repair
Power boats	100	0.3	300	10,000
Sailboats	300	0.1	1000	400,000
Luxury yachts	50	0.6	5000	2,000,000

At most one repair is required per boat each year. Repair incidence and cost are mutually independent. The marina budgets an amount, [math]Y[/math], equal to the aggregate mean repair costs plus the standard deviation of the aggregate repair costs.

Calculate [math]Y[/math].

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

Key: B

The number of repairs for each boat type has a binomial distribution. For power boats:

[[math]] \begin{aligned} &\operatorname{E}[ S ] = 100(0.3)(300) = 9, 000, \\ &\operatorname{E}[ S ] = 100(0.3)10, 000) + 100(0.3)(0.7)(300 2 ) = 2,190, 000 \end{aligned} [[/math]]

For sail boats:

[[math]] \begin{aligned} &\operatorname{E}[ S ] = 300(0.1)(1, 000) = 30, 000, \\ &\operatorname{E}[ S ] = 300(0.1)(400, 000) + 300(0.1)(0.9)(1, 0002 ) = 39, 000, 000 \end{aligned} [[/math]]

For luxury yachts:

[[math]] \begin{aligned} &\operatorname{E}[ S ] = 50(0.6)(5, 000) = 150, 000, \\ &\operatorname{E}[ S ] = 50(0.6)(0.4)(2, 000, 000) + 50(0.6)(0.4)(5, 0002 ) = 360, 000, 000 \end{aligned} [[/math]]

The sums are 189,000 expected and a variance of 401,190,000 for a standard deviation of 20,030. The mean plus standard deviation is 209,030.

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