An insurance company pays out claims on its life insurance...

An insurance company pays out claims on its life insurance policies in accordance with a Poisson process having rate $\lambda=5$ per week. If the amount of money paid on each policy is exponentially distributed with mean $\$ 2000$, what is the mean and variance of the amount of money paid by the insurance company in a four-week span?

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Key Concepts

Poisson Process

A Poisson process is a stochastic process that models the occurrence of events randomly over time, where events occur independently and at a constant average rate. The number of events occurring in a fixed time interval follows a Poisson distribution, characterized by a parameter that is the product of the rate and the length of the interval.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between independent events. It is characterized by its memorylessness property and has both its mean and variance directly related to its rate parameter, with the variance being the square of the mean.

Compound Poisson Process

A compound Poisson process is constructed by summing a random number of independent identically distributed random variables, where the number of terms in the sum is governed by a Poisson process. This model is used to represent aggregated totals, such as total claim amounts, and its mean and variance can be derived using the law of total expectation and the law of total variance, which account for both the randomness in the number of events and the randomness in the size of each event.

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