An actuary is studying hurricane models. A year is classified as a high, medium, or low hurricane year with probabilities 0.1, 0.3, and 0.6, respectively. The numbers of hurricanes in high, medium, and low years follow Poisson distributions with means 20, 15, and 10, respectively.

Calculate the variance of the number of hurricanes in a randomly selected year.

• 11.25
• 12.50
• 12.94
• 13.42
• 23.75

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

Step 1: Understanding the Problem Setup

Define [math]N[/math] as the number of hurricanes in a randomly selected year and [math]Y[/math] as the type of year (High, Medium, or Low). The problem provides the following information regarding the hurricane year types and their associated Poisson distribution parameters:

Problem Parameters

Year Type ([math]Y[/math])	Probability ([math]P(Y)[/math])	Poisson Mean ([math]\lambda = E[N|Y][/math])
High	[math]0.1[/math]	[math]20[/math]
Medium	[math]0.3[/math]	[math]15[/math]
Low	[math]0.6[/math]	[math]10[/math]

For a Poisson distribution with parameter [math]\lambda[/math], the following properties are essential for this problem:

• The expected value is [math]E[X] = \lambda[/math].
• The variance is [math]\text{Var}[X] = \lambda[/math].
• The second moment, [math]E[X^2][/math], which is useful for calculating variance, is given by [math]E[X^2] = \text{Var}[X] + (E[X])^2 = \lambda + \lambda^2[/math].

Step 2: Calculate the Expected Number of Hurricanes (First Moment)

The expected number of hurricanes in a randomly selected year, [math]E[N][/math], is calculated using the Law of Total Expectation. This law states that the overall expected value can be found by taking the weighted average of the conditional expected values:

[[math]]E[N] = E[E[N|Y]] = \sum_{y \in \{ \text{High, Medium, Low} \}} E[N|Y=y]P(Y=y)[[/math]]

Substituting the given probabilities and conditional means from the problem setup:

[[math]]E[N] = (20)(0.1) + (15)(0.3) + (10)(0.6)[[/math]]

[[math]]E[N] = 2.0 + 4.5 + 6.0[[/math]]

[[math]]E[N] = 12.5[[/math]]

Step 3: Calculate the Second Moment of the Number of Hurricanes

To find the variance of [math]N[/math], we also need to calculate its second moment, [math]E[N^2][/math]. First, we determine the conditional second moment for each year type. For a Poisson distribution with parameter [math]\lambda[/math], the second moment is [math]E[X^2] = \lambda + \lambda^2[/math]. Let's compute [math]E[N^2|Y=y][/math] for each year type:

Conditional Second Moments for Each Year Type

Year Type ([math]Y[/math])	Poisson Mean ([math]\lambda[/math])	[math]\lambda^2[/math]	Conditional Second Moment ([math]E[N^2|Y] = \lambda + \lambda^2[/math])
High	[math]20[/math]	[math]20^2 = 400[/math]	[math]20 + 400 = 420[/math]
Medium	[math]15[/math]	[math]15^2 = 225[/math]	[math]15 + 225 = 240[/math]
Low	[math]10[/math]	[math]10^2 = 100[/math]	[math]10 + 100 = 110[/math]

Next, we apply the Law of Total Expectation to find the overall second moment [math]E[N^2][/math]:

[[math]]E[N^2] = E[E[N^2|Y]] = \sum_{y \in \{ \text{High, Medium, Low} \}} E[N^2|Y=y]P(Y=y)[[/math]]

Substituting the calculated conditional second moments and probabilities:

[[math]]E[N^2] = (420)(0.1) + (240)(0.3) + (110)(0.6)[[/math]]

[[math]]E[N^2] = 42.0 + 72.0 + 66.0[[/math]]

[[math]]E[N^2] = 180[[/math]]

Step 4: Calculate the Variance of the Number of Hurricanes

Finally, with both the first moment ([math]E[N][/math]) and the second moment ([math]E[N^2][/math]) calculated, we can determine the variance of the number of hurricanes, [math]\text{Var}[N][/math], using the fundamental formula:

[[math]]\text{Var}[N] = E[N^2] - (E[N])^2[[/math]]

Substitute the values obtained in the previous steps:

[[math]]\text{Var}[N] = 180 - (12.5)^2[[/math]]

[[math]]\text{Var}[N] = 180 - 156.25[[/math]]

[[math]]\text{Var}[N] = 23.75[[/math]]

Key Insights

• Law of Total Expectation: This fundamental principle allows calculating the overall expected value of a random variable by conditioning on another variable. It is expressed as [math]E[X] = E[E[X|Y]][/math] or [math]\sum P(Y=y)E[X|Y=y][/math].
• Second Moment Calculation: The second moment [math]E[X^2][/math] can also be found using the Law of Total Expectation, [math]E[X^2] = E[E[X^2|Y]][/math], which is crucial for calculating variance.
• Poisson Distribution Properties: For a Poisson random variable with mean [math]\lambda[/math], its expected value is [math]E[X] = \lambda[/math], its variance is [math]\text{Var}[X] = \lambda[/math], and its second moment is [math]E[X^2] = \lambda + \lambda^2[/math]. These properties simplify calculations in mixed Poisson models.
• Variance from Moments: The variance of any random variable can always be calculated as the difference between its second moment and the square of its first moment: [math]\text{Var}[X] = E[X^2] - (E[X])^2[/math]. This is a widely applicable formula.
• Law of Total Variance (Alternative Approach): Although not explicitly used in this solution's step-by-step breakdown, an alternative and equally valid approach is the Law of Total Variance: [math]\text{Var}[X] = E[\text{Var}[X|Y]] + \text{Var}[E[X|Y]][/math]. This method directly decomposes the variance.

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Solution: E

The mean is the weighted average of the three means: 0.1(20) + 0.3(15) + 0.6(10) = 12.5. The second moment is the weighted average of the three second moments (each of which is the square of the mean plus the mean, for a Poisson distribution): 0.1(420) + 0.3(240) + 0.6(110) = 180. The variance is the second moment minus the square of the mean, which is 23.75.

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