# What Is the Variance of the Number of Claims Filed in a Poisson Distribution?

• buzzmath
In summary, the actuary has discovered that policyholders are three times as likely to file two claims as to file four claims, with a Poisson distribution. To find the variance of the number of claims filed, the equation P(2) = 3P(4) is used and simplified to 4\lambda^2 = \lambda^4.
buzzmath

## Homework Statement

An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?

2. Homework Equations [/]
P(x)=e^(-lamda)*(lamda^x)/x! Var(x)=lamda

## The Attempt at a Solution

I know that P(2)=3P(4) and that P(2)=e^(-lamda)*(lamda^2)/2 and that P(4)=e^(-lamda)*(lamda^4)/4! So I set these equal to find lamda and that is the variance. The solution given to me is P(2)=e^(lamda)*(lamda^2)/2=3*e^(-lamda)*(lamda^4)/4!=3*P(4)*24*lamda^2=6*lamda^4 I don't understand the last two equalities. Any help? where do the 24*lamda^2 comefrom and how do they get 6*lamda^4?
Thanks

Is this a typo? if so how would I go about solving the problem using the method I've started? or any other way?
Thanks

From the given information,
$$P(2) = \frac{e^{-\lambda}\lambda^2}{2!} = 3P(4) = 3\frac{e^{-\lambda}\lambda^4}{4!}$$
From this, we get
$$\frac{e^{-\lambda}\lambda^2}{2} = 3\frac{e^{-\lambda}\lambda^4}{24}$$
so
$$\frac{e^{-\lambda}\lambda^2}{2} = \frac{e^{-\lambda}\lambda^4}{8}$$
or
$$4e^{-\lambda}\lambda^2} = e^{-\lambda}\lambda^4}$$
That works out to $4\lambda^2 = \lambda^4$
Can you take it from there?

BTW, the name of this Greek letter is lambda.

## What is the Poisson Distribution?

The Poisson Distribution is a probability distribution that is used to model the probability of a certain number of events occurring within a fixed time interval, given the average number of events that occur in that interval.

## How is the Poisson Distribution different from other probability distributions?

The Poisson Distribution is characterized by only one parameter, the average number of events, whereas other distributions may have multiple parameters. Additionally, the Poisson Distribution is used to model the probability of discrete events, rather than continuous events.

## When should the Poisson Distribution be used?

The Poisson Distribution is best used when the following conditions are met: the events are independent, the average number of events is known, events occur at a constant rate, and the events are finite (meaning they cannot occur an infinite number of times).

## How is the Poisson Distribution calculated?

The formula for the Poisson Distribution is P(x; μ) = (e^-μ * μ^x) / x!, where x is the number of events, and μ is the average number of events. This formula gives the probability of x events occurring in a given interval.

## What are some applications of the Poisson Distribution?

The Poisson Distribution has many practical applications, such as modeling the number of customers arriving at a store, the number of accidents on a highway, or the number of phone calls received at a call center. It can also be used in fields such as biology, epidemiology, and finance.

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