I have a small problem regarding to expectation on second moment. It would be lovely if you guys can give me a hand.
The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of $d$, the insurance comp reduces the expected claim payment by $10\%$.
Calculate the percentage reduction on the variance of the claim payment.
First I make $X$ the original payment and $Y$ the payment with deductible. It's pretty obvious that $X$ has $E(X)=\lambda$ and its variance is simply $ \lambda^2$ and $E(X^2)$ becomes $2\lambda^2$.
Since its just $90\%$ then $E(Y)$ becomes $0.9E(X)=0.9\lambda$ but I need $E(Y^2)$ in order to find the $\operatorname{Var}(Y)$. Could anyone help me with this? Thanks.
I think I see where your doubt is. I guess the question boils down to: If $\mathbb E[Y]=a\mathbb E[X]$ for some const $a$ and both $X$ and $Y$ are exponentially distributed, does this mean $Y=aX$ in distribution? The answer to that is yes:
You have $$ \mathbb E[Y]=a\mathbb E[X] \Leftrightarrow \mathbb E[Y]=\mathbb E[aX] $$
So $Y$ and $aX$ have the same mean. But since the exponential distribution is characterised by this single parameter, they have to have the same distribution.