Claim amounts are independent random variables with probability density function

[[math]] f(x) = \begin{cases} \frac{10}{x^2}, \, x \gt10 \\ 0, \, \textrm{otherwise} \end{cases} [[/math]]

Calculate the probability that the largest of three randomly selected claims is less than 25.

• 8/125
• 12/125
• 27/125
• 2/5
• 3/5

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

Step 1: Understand the Problem and Probability Density Function

The problem asks for the probability that the largest of three independent claims is less than 25. This means that all three claims must individually be less than 25. Let [math]X_1, X_2, X_3[/math] be the three independent claim amounts. We are looking for [math]P(\max(X_1, X_2, X_3) \lt 25)[/math]. This is equivalent to [math]P(X_1 \lt 25 \text{ and } X_2 \lt 25 \text{ and } X_3 \lt 25)[/math]. Due to the independence of the claims, this can be written as [math]P(X \lt 25)^3[/math]. The probability density function (PDF) for a single claim amount [math]X[/math] is given by:

[[math]] f(x) = \begin{cases} \frac{10}{x^2}, \, x \gt10 \\ 0, \, \textrm{otherwise} \end{cases} [[/math]]

Step 2: Calculate the Cumulative Distribution Function (CDF) for a Single Claim

To find the probability that a single claim [math]X[/math] is less than or equal to a value [math]x[/math] (where [math]x \gt 10[/math]), we need to integrate the PDF from the lower bound of the distribution (10) to [math]x[/math]. This gives us the cumulative distribution function (CDF), [math]F(x) = P(X \le x)[/math].

[[math]] F(x) = \operatorname{P}( X \le x) = \int_{10}^{x} \frac{10}{t^2} dt [[/math]]

Now, we perform the integration:

[[math]] \int_{10}^{x} 10t^{-2} dt = 10 \left[ \frac{t^{-1}}{-1} \right]_{10}^{x} = 10 \left[ -\frac{1}{t} \right]_{10}^{x} [[/math]]

Evaluating the definite integral:

[[math]] F(x) = 10 \left( -\frac{1}{x} - \left(-\frac{1}{10}\right) \right) = 10 \left( -\frac{1}{x} + \frac{1}{10} \right) = -\frac{10}{x} + \frac{10}{10} = 1 - \frac{10}{x} [[/math]]

So, the CDF for [math]x \gt 10[/math] is [math]F(x) = 1 - \frac{10}{x}[/math].

Step 3: Determine the Probability for a Single Claim to be Less Than 25

Using the CDF derived in Step 2, we can now find the probability that a single claim [math]X[/math] is less than 25. We set [math]x = 25[/math]:

[[math]] \operatorname{P}( X \le 25) = F(25) = 1 - \frac{10}{25} = 1 - \frac{2}{5} = \frac{3}{5} [[/math]]

Thus, the probability that any single claim is less than 25 is [math]\frac{3}{5}[/math].

Step 4: Calculate the Probability that the Largest of Three Claims is Less Than 25

As established in Step 1, the probability that the largest of three claims is less than 25 is equivalent to the probability that all three claims are individually less than 25. Since the claims are independent, we can multiply their individual probabilities:

[[math]] \operatorname{P}(\max(X_1, X_2, X_3) \lt 25) = \operatorname{P}(X_1 \lt 25) \times \operatorname{P}(X_2 \lt 25) \times \operatorname{P}(X_3 \lt 25) [[/math]]

Given that [math]\operatorname{P}(X \lt 25) = \frac{3}{5}[/math], we have:

[[math]] \operatorname{P}(\max(X_1, X_2, X_3) \lt 25) = \left(\frac{3}{5}\right)^3 = \frac{3^3}{5^3} = \frac{27}{125} [[/math]]

The probability that the largest of three randomly selected claims is less than 25 is [math]\frac{27}{125}[/math].

Key Insights

• The phrase "largest of [math]n[/math] random variables is less than [math]x[/math]" implies that all [math]n[/math] random variables are less than [math]x[/math].
• For independent random variables, the probability of multiple events occurring simultaneously is the product of their individual probabilities.
• To calculate the probability [math]P(X \le x)[/math] from a given probability density function [math]f(x)[/math], integrate [math]f(x)[/math] from its lower bound up to [math]x[/math] to find the Cumulative Distribution Function (CDF).
• Understanding the bounds of the PDF is crucial for setting up the integral correctly.

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

Solution: C

The probability that a single claim is less than 25 is

[[math]] \operatorname{P}( X \leq 25) = \int_{10}^{25} 10x^{-2} dx = 3/5. [[/math]]

The probability that all three claims are less than 25 is (3 / 5)³ = 27 /125.

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