I have what appears to be a relatively simple question, but am struggling to understand how to go about answering it.
The general question is as follows:
What is the expected value of $S_{I}$, where:
$S_{I} = S$ if $S <3000 $
$S_{I} = 3000$ if $S >3000$
where $S$ is a compound distribution (details not necessary for my problem here)
My initial attempt is as follows:
$E[S_{I}] = E[E[S_{I}|S]] = E[S * P(S\leq3000) + 3000*P(S>3000)] = E[S]*P(S\leq 3000)+3000*P(S>3000).$
Now, from the definition of $S_{I}$, it is clear that we should have $E[S_{I}]<3000$. For this particular problem $E[S] = 4000$, and hence my proposed method for solving is wrong.
So I thought that my $E[S]$ in the line above should actually be $E[S|S\leq 3000]$
Is this assumption correct?
If so, is it true that $E[S|S\leq 3000] = E[S]*P(S\leq 3000)$ ??
Thanks for any help.
Let $x=3000$, then $$ E(S_{I})=E(S;S\leqslant x)+x\,P(S\gt x), $$ and $$ E(S_{I})=E(S\mid S\leqslant x)\,P(S\leqslant x)+x\,P(S\gt x). $$