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:

$E[S_{I}] = E[E[S_{I}|S]] = E[S * P(Sleq3000) + 3000*P(S>3000)] = E[S]*P(Sleq 3000)+3000*P(S>3000).$

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 $E[S|Sleq 3000]$

Is this assumption correct?

If so, is it true that $E[S|Sleq 3000] = E[S]*P(Sleq 3000)$ ??

Thanks for any help.

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#### Best Answer

Assuming all expectations are defined, by the law of total expectation, $$E[S]=E(S,|,Sleq 3000),P(Sleq 3000) + E(S,|,S > 3000),P(S> 3000),,$$ and from it $$E[S_I]=E(S,|,Sleq 3000),P(Sleq 3000) + 3000,P(S> 3000),,$$ as you stated. But it is not true in general that $E(S,|,Sleq 3000)=E(S),P(Sleq 3000)$. To see why, imagine an $S$ that takes either the value 0 or the value 4000 each with probability 1/2. Then $E(S)P(Sleq 3000)=2000cdot1/2$ and $E(S,|,Sleq 3000)=0$.

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