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An insurance company receives a certain number of claims of week that are Poisson distributed at a $\text{rate} = 10$. The amount of each claim is exponentially distributed with $\text{mean} = \$1,\!000$.

Calculate:

a) The mean and variance of the amount paid out by the company in a six week span.

b) The probability that the amount paid out in a six week span is greater than $\$15,\!000$.

How I think it can be solved:

I treated it like a compound random variable ( poisson x exponential) and hence determine the probability distribution of the amount paid per week. Then extend that to six weeks and use standard mean/ variance formulas to calculate. My issue is that not sure how to calculate the correct compound random variable.

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StatVoyager
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  • Welcome to MathSE. This question reads like a homework problem that you have been assigned. What have you done yourself to answer these questions? You need to include some work of your own (have a question other than the math problem you have been given to solve), else this will be closed. – Mike Pierce Jul 10 '15 at 15:04

1 Answers1

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I don´t really know, where you stuck. I´m starting with the variance. Let X be be the poisson distributed r.v. and Y the exponentially distributed r.v. Then you have to calculate $Var(X\cdot Y)$.

$Var(X\cdot Y)=E[(XY)^2]-(E[XY])^2$

X and Y are independent. Therefore $(E[XY])^2=(E[X]\cdot E[Y])^2$

$Var(X\cdot Y)=E[(XY)^2]-(E[XY])^2=E[(XY)^2]-(E[X]\cdot E[Y])^2$

$=E(X^2)\cdot E(Y^2)-(E[X]\cdot E[Y])^2$

$=(Var[X]+E^2[X])\cdot (Var[Y]+E^2[Y])-E^2[X]\cdot E^2[Y]$

$=Var[X]\cdot Var[Y]+Var[X]\cdot E^2[Y] + Var[Y]\cdot E^2[X]+E^2[X]\cdot E^2[Y]-E^2[X]\cdot E^2[Y]$

$Var(X\cdot Y)=Var[X]\cdot Var[Y]+Var[X]\cdot E^2[Y] + Var[Y]\cdot E^2[X]$

callculus42
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