# Finding out the probability density function

We let $X$ as the avtual claim amount , $Y$ is the amount paid by the insurer. The insurer takes up an excess of loss reinsurance arrangement with retention limit $M$ , we need to find out the probability density function of $Y$ ( $f_{Y}(y)$ ) , in terms of $f_{X}(x)$.

The random variable $Y$ is defined as :

$$Y = \begin{cases} X & ,X\:<\:M \\ M & ,X\:\geq\: M \\ \end{cases}$$

So I tried as follows :

I started with $P(Y\leq y)$ , for $X \: <\:M$,

$$P(Y\leq y)=P(X\leq y)=\int^y_0 f(x)dx,$$

Thus differentiating gives $f_{Y}(y) = f_{X}(x)$ for $X\:<\:M$.

I also tried for the $X\geq M$ part but couldn't conclude anything , it went like this :

• $P(Y\leq y)\:=\:P(M\leq y)$ , I got stuck here , since $M$ is a constant , how to proceed ? Can anyone tell ?
• Or is there a better way to solve these kind of questions ?

Work directly with the density function for the $Y \geq M$ case. You have a censored distribution, with $f_Y(M) = 1 - F(X < M)$ & $\,f_Y(y) = 0\, \forall\, y > M$. Note that in the case that $X$ is a continuous distribution, $F(X < M) = F(X \leq M)$.