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