In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be mutually independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4.

Calculate the probability that the maximum of these losses exceeds 3.

• 0.002
• 0.050
• 0.159
• 0.287
• 0.414

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

Step 1: Define Random Variables and the Target Probability

Let the random variables representing the annual losses due to storm, fire, and theft be [math]S, F,[/math] and [math]T[/math], respectively. These losses are assumed to be mutually independent and exponentially distributed with the following given means:

• Mean of Storm Loss ([math]S[/math]): [math]\mu_S = 1.0[/math]
• Mean of Fire Loss ([math]F[/math]): [math]\mu_F = 1.5[/math]
• Mean of Theft Loss ([math]T[/math]): [math]\mu_T = 2.4[/math]

We are interested in the maximum of these losses. Let [math]Y[/math] be the maximum loss, so [math]Y = \max(S, F, T)[/math]. The objective is to calculate the probability that the maximum of these losses exceeds 3, which is [math]\operatorname{P}[Y \gt 3][/math].

Step 2: Use the Complement Rule and Independence

To find [math]\operatorname{P}[Y \gt 3][/math], it is often simpler to calculate the probability of its complement, [math]\operatorname{P}[Y \le 3][/math], and subtract it from 1.

[[math]]\operatorname{P}[Y \gt 3] = 1 - \operatorname{P}[Y \le 3][[/math]]

The event [math]Y \le 3[/math] means that the maximum of the three losses is less than or equal to 3. This condition implies that each individual loss must be less than or equal to 3 simultaneously.

[[math]]\operatorname{P}[\max(S, F, T) \le 3] = \operatorname{P}[S \le 3 \text{ and } F \le 3 \text{ and } T \le 3][[/math]]

Given that the losses [math]S, F, T[/math] are mutually independent, the joint probability can be factored into the product of individual probabilities:

[[math]]\operatorname{P}[S \le 3 \text{ and } F \le 3 \text{ and } T \le 3] = \operatorname{P}[S \le 3] \cdot \operatorname{P}[F \le 3] \cdot \operatorname{P}[T \le 3][[/math]]

Combining these, the target probability can be expressed as:

[[math]]\operatorname{P}[Y \gt 3] = 1 - \left( \operatorname{P}[S \le 3] \cdot \operatorname{P}[F \le 3] \cdot \operatorname{P}[T \le 3] \right)[[/math]]

Step 3: Calculate Individual Probabilities Using Exponential CDF

For an exponentially distributed random variable [math]X[/math] with mean [math]\mu[/math], the rate parameter is [math]\lambda = 1/\mu[/math]. The Cumulative Distribution Function (CDF) for [math]X[/math] is given by:

[[math]]F_X(x) = \operatorname{P}[X \le x] = 1 - e^{-\lambda x}[[/math]]

We will calculate [math]\operatorname{P}[\text{Loss} \le 3][/math] for each type of loss using their respective means:

Individual Probabilities for Losses

Loss Type	Mean ([math]\mu[/math])	Rate ([math]\lambda = 1/\mu[/math])	[math]\operatorname{P}[\text{Loss} \le 3] = 1 - e^{-\lambda \cdot 3}[/math]
Storm (S)	[math]1.0[/math]	[math]\lambda_S = 1/1.0 = 1[/math]	[math]\operatorname{P}[S \le 3] = 1 - e^{-1 \cdot 3} = 1 - e^{-3}[/math]
Fire (F)	[math]1.5[/math]	[math]\lambda_F = 1/1.5 = 2/3[/math]	[math]\operatorname{P}[F \le 3] = 1 - e^{-(2/3) \cdot 3} = 1 - e^{-2}[/math]
Theft (T)	[math]2.4[/math]	[math]\lambda_T = 1/2.4 = 5/12[/math]	[math]\operatorname{P}[T \le 3] = 1 - e^{-(5/12) \cdot 3} = 1 - e^{-5/4} = 1 - e^{-1.25}[/math]

Step 4: Compute the Final Probability

Now, substitute the individual probabilities calculated in Step 3 back into the expression from Step 2:

[[math]]\operatorname{P}[Y \gt 3] = 1 - \left( (1 - e^{-3}) \cdot (1 - e^{-2}) \cdot (1 - e^{-1.25}) \right)[[/math]]

Using numerical approximations for the exponential terms:

• [math]e^{-3} \approx 0.049787[/math]
• [math]e^{-2} \approx 0.135335[/math]
• [math]e^{-1.25} \approx 0.286505[/math]

Calculate the individual probabilities:

• [math]\operatorname{P}[S \le 3] = 1 - 0.049787 = 0.950213[/math]
• [math]\operatorname{P}[F \le 3] = 1 - 0.135335 = 0.864665[/math]
• [math]\operatorname{P}[T \le 3] = 1 - 0.286505 = 0.713495[/math]

Now, substitute these into the formula for [math]\operatorname{P}[Y \gt 3][/math]:

[[math]]\operatorname{P}[Y \gt 3] = 1 - (0.950213 \cdot 0.864665 \cdot 0.713495)[[/math]]

First, multiply the probabilities:

[[math]]0.950213 \cdot 0.864665 \approx 0.821033[[/math]]

Then, continue the multiplication:

[[math]]0.821033 \cdot 0.713495 \approx 0.585520[[/math]]

Finally, calculate the desired probability:

[[math]]\operatorname{P}[Y \gt 3] \approx 1 - 0.585520 \approx 0.41448[[/math]]

Rounding to three decimal places, the probability is approximately [math]0.414[/math].

Key Insights

• When dealing with the probability of the maximum of several independent random variables being less than a certain value, you can multiply their individual probabilities of being less than that value: [math]\operatorname{P}[\max(X_1, \dots, X_n) \le k] = \prod_{i=1}^n \operatorname{P}[X_i \le k][/math].
• The complement rule ([math]\operatorname{P}[A] = 1 - \operatorname{P}[A^c][/math]) is a powerful tool, especially when calculating probabilities for events like "exceeds" or "at least", as it often simplifies the calculation.
• For an exponential distribution, remember the relationship between the mean [math]\mu[/math] and the rate parameter [math]\lambda[/math]: [math]\lambda = 1/\mu[/math]. The Cumulative Distribution Function (CDF) for an exponential variable [math]X[/math] is [math]F_X(x) = \operatorname{P}[X \le x] = 1 - e^{-\lambda x}[/math].

This article was generated by AI and may contain errors. If permitted, please edit the article to improve it.

Solution: E

Let [math]S, F,[/math] and [math]T[/math] be the losses due to storm, fire, and theft respectively. Let [math]Y = \max(S,F,T)[/math]. Then,

[[math]] \begin{align*} \operatorname{P}[Y \gt 3] =1 − \operatorname{P}[Y ≤ 3] =1 − \operatorname{P}[\max( S , F , T ) ≤ 3] &=1 − \operatorname{P}[ S ≤ 3]\operatorname{P}[ F ≤ 3]\operatorname{P}[T ≤ 3]\\ &= 1 − (1 − e^{−3/1} )(1 − e^{−3/1.5} )(1 − e ^{-3/2.4} ) \\ &=0.414. \end{align*} [[/math]]

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