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

Step 1: Identify the Conditional Probability Distribution

First, we identify the joint probabilities for the annual number of tornadoes in county Q, given that there are no tornadoes in county P (i.e., [math]P=0[/math]). We extract the row corresponding to [math]P=0[/math] from the given joint distribution table:

Q=0	Q=1	Q=2	Q=3
0.12	0.06	0.05	0.02

Next, we calculate the marginal probability of [math]P=0[/math], which will be the denominator for our conditional probabilities:

[[math]]P(P=0) = P(Q=0, P=0) + P(Q=1, P=0) + P(Q=2, P=0) + P(Q=3, P=0)[[/math]]

[[math]]P(P=0) = 0.12 + 0.06 + 0.05 + 0.02 = 0.25[[/math]]

Now, we determine the conditional probability distribution [math]P(Q=q | P=0)[/math] using the formula [math]P(Q=q | P=0) = \frac{P(Q=q, P=0)}{P(P=0)}[/math].

[math]q[/math]	[math]P(Q=q, P=0)[/math]	[math]P(Q=q | P=0)[/math]
0	0.12	[math]\frac{0.12}{0.25} = \frac{12}{25}[/math]
1	0.06	[math]\frac{0.06}{0.25} = \frac{6}{25}[/math]
2	0.05	[math]\frac{0.05}{0.25} = \frac{5}{25}[/math]
3	0.02	[math]\frac{0.02}{0.25} = \frac{2}{25}[/math]

Step 2: Calculate the Conditional Expectation (Mean)

The conditional expectation, or mean, of [math]Q[/math] given [math]P=0[/math] is calculated using the formula [math]E[Q | P=0] = \sum_{q} q \cdot P(Q=q | P=0)[/math]:

[[math]]E[Q | P=0] = \left(0 \cdot \frac{12}{25}\right) + \left(1 \cdot \frac{6}{25}\right) + \left(2 \cdot \frac{5}{25}\right) + \left(3 \cdot \frac{2}{25}\right)[[/math]]

[[math]]E[Q | P=0] = \frac{0 + 6 + 10 + 6}{25} = \frac{22}{25}[[/math]]

[[math]]E[Q | P=0] = 0.88[[/math]]

Step 3: Calculate the Conditional Second Moment

The conditional second moment of [math]Q[/math] given [math]P=0[/math] is calculated using the formula [math]E[Q^2 | P=0] = \sum_{q} q^2 \cdot P(Q=q | P=0)[/math]:

[[math]]E[Q^2 | P=0] = \left(0^2 \cdot \frac{12}{25}\right) + \left(1^2 \cdot \frac{6}{25}\right) + \left(2^2 \cdot \frac{5}{25}\right) + \left(3^2 \cdot \frac{2}{25}\right)[[/math]]

[[math]]E[Q^2 | P=0] = \frac{(0 \cdot 12) + (1 \cdot 6) + (4 \cdot 5) + (9 \cdot 2)}{25}[[/math]]

[[math]]E[Q^2 | P=0] = \frac{0 + 6 + 20 + 18}{25} = \frac{44}{25}[[/math]]

[[math]]E[Q^2 | P=0] = 1.76[[/math]]

Step 4: Calculate the Conditional Variance

The conditional variance of [math]Q[/math] given [math]P=0[/math] is calculated using the formula [math]Var[Q | P=0] = E[Q^2 | P=0] - (E[Q | P=0])^2[/math]. Using the results from the previous steps:

[[math]]Var[Q | P=0] = \frac{44}{25} - \left(\frac{22}{25}\right)^2[[/math]]

[[math]]Var[Q | P=0] = \frac{44}{25} - \frac{484}{625}[[/math]]

To combine these fractions, we find a common denominator, which is 625:

[[math]]Var[Q | P=0] = \frac{44 \cdot 25}{25 \cdot 25} - \frac{484}{625}[[/math]]

[[math]]Var[Q | P=0] = \frac{1100}{625} - \frac{484}{625}[[/math]]

[[math]]Var[Q | P=0] = \frac{1100 - 484}{625} = \frac{616}{625}[[/math]]

[[math]]Var[Q | P=0] = 0.9856[[/math]]

Rounding to two decimal places, the conditional variance is approximately 0.99.

Key Insights

• Conditional Probability Distribution: The first step in calculating any conditional moment is to accurately determine the conditional probability distribution. This involves dividing the joint probabilities of the specific condition by the marginal probability of that condition.
• Marginal Probability: The marginal probability of the conditioning event (e.g., [math]P(P=0)[/math]) serves as the denominator for all conditional probabilities within that event, ensuring the conditional probabilities sum to 1.
• Variance Calculation: The variance can be efficiently calculated using the formula [math]Var(X) = E[X^2] - (E[X])^2[/math], requiring the computation of both the first moment (mean) and the second moment of the random variable.
• Precision: Maintaining fractions in intermediate calculations can help preserve precision, especially when dealing with exact values, before rounding the final result.

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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.