Let [math]N[/math] denote the number of accidents occurring during one month on the northbound side of a highway and let [math]S[/math] denote the number occurring on the southbound side. Suppose that [math]N[/math] and [math]S[/math] are jointly distributed as indicated in the table.

N\S	0	1	2	3 or more
0	0.04	0.06	0.10	0.04
1	0.10	0.18	0.08	0.03
2	0.12	0.06	0.05	0.02
3 or more	0.05	0.04	0.02	0.01

Calculate [math]\operatorname{Var}(N | N + S = 2) [/math].

• 0.48
• 0.55
• 0.67
• 0.91
• 1.25

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

Solution: B

Given N + S = 2, there are 3 possibilities (N,S) = (2,0), (1,1), (0,2) with probabilities 0.12, 0.18, and 0.10 respectively. The associated conditional probabilities are

[[math]] \begin{align*} P(N = 0 | N + S = 2) = 0.10/0.40 &= 0.25, \\ P(N = 1 | N + S = 2) = 0.18/0.40 &= 0.45, \\ P(N = 2 | N + S = 2) = 0.12/0.40 &= 0.30. \\ \end{align*} [[/math]]

The mean is 0.25(0) + 0.45(1) + 0.30(2) = 1.05. The second moment is 0.25(0) + 0.45(1) + 0.30(4) = 1.65. The variance is 1.65 – (1.05)(1.05) = 0.5475.

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