An actuary determines that the annual number of tornadoes in counties P and Q are jointly distributed as follows:

	Q= 0	Q=1	Q=2	Q=3
P=0	0.12	0.06	0.05	0.02
P=1	0.13	0.15	0.12	0.03
P=2	0.05	0.15	0.10	0.02

Calculate the conditional variance of the annual number of tornadoes in county Q, given that there are no tornadoes in county P.

• 0.51
• 0.84
• 0.88
• 0.99
• 1.76

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

Solution: D

With no tornadoes in County P the probabilities of 0, 1, 2, and 3 tornadoes in County Q are 12/25, 6/25, 5/25, and 2/25 respectively. The mean is (0 + 6 + 10 + 6)/25 = 22/25. The second moment is (0 + 6 + 20 + 18)/25 = 44/25. The variance is 44/25 – (22/25)2 = 0.9856.

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