Insurance Claims An insurance company's monthly claims are...

Insurance Claims An insurance company's monthly claims are modeled by a continuous, positive random variable $X,$ whose probability density function is proportional to $(1+x)^{-4}$ where $0<$x$<\infty .$ Determine the company's expected monthly claims. Choose one of the following. Source: Society of Actuaries. $\begin{array}{ll}{\text { (a) } 1 / 6} & {\text { (b) } 1 / 3} & {\text { (c) } 1 / 2} & {\text { (d) } 1} & {\text { (e) } 3}\end{array}$

Insurance Claims An insurance company's monthly claims are modeled by a continuous, positive random variable $X,$ whose probability density function is proportional to $(1+x)^{-4}$ where $0<$x$<\infty .$ Determine the company's expected monthly claims. Choose one of the following. Source: Society of Actuaries.
$\begin{array}{ll}{\text { (a) } 1 / 6} & {\text { (b) } 1 / 3} & {\text { (c) } 1 / 2} & {\text { (d) } 1} & {\text { (e) } 3}\end{array}$

Key Concepts

Expected Value of a Continuous Random Variable

The expected value, or mean, of a continuous random variable is a measure of the central tendency of its distribution and is calculated by integrating the product of the variable and its PDF over the entirety of its support. It provides a long-run average outcome if the experiment were repeated infinitely often.

Improper Integrals and Convergence

Many probability distributions, especially those defined on an infinite interval, require the evaluation of improper integrals. These integrals must converge (yield finite values) in order for quantities like the normalization constant and the expected value to be well-defined. The techniques employed often involve substitutions or the use of known integral formulas, such as those related to the gamma or beta functions.

Probability Density Function (PDF)

A probability density function (PDF) is a function that describes the relative likelihood for a continuous random variable to take on a given value. It must be nonnegative over its domain and integrate to one, ensuring that the total probability is correctly normalized over the space of outcomes.

Normalization of a PDF

When a function is said to be 'proportional to' a given form, it means there exists a constant factor that scales it into a proper PDF. Finding this normalization constant requires solving an integral over the support of the random variable such that the resulting function integrates to one. This process is fundamental for using the function in probability calculations.

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