Computing $E[X 1_{ \{ |Y|<1 \}}]$

Question

Am looking for any suggestions on how to compute \begin{align*} E[X \ 1_{ \{ |Y|<1 \}}] \end{align*}

where $X$ and $Y$ are finite variance random variable assume zero mean.

In my setting $Y=X+W$ where $X$ and $W$ are independent.

Answer 1

This is a somewhat open question, but I will give one possible approach.

My suggestion is to use the law of iterated expectations:

$$E[X1_{\{|Y|<1\}}] = E[E[X1_{\{|Y|<1\}}|Y]]=E[1_{\{|Y|<1\}}E[X|Y]],$$

where I condition on $Y$ and can move the indicator function outside of the inner expectation, since $1_{\{|Y|<1\}}$ is $Y$-measurable. Now notice that you need only calculate the expectation of $X$ conditional on $Y$, say $g(Y)=E[X|Y]$, and you are left with the expectation $E[1_{\{|Y|<1\}}g(Y)]$ involving only the random variable $Y$, which is given by $$E[1_{\{|Y|<1\}}g(Y)] = \int 1_{\{|y|<1\}}g(y)\text{d}F_Y(y)=\int_{-1}^1 g(y)\text{d}F_Y(y),$$ where $F_Y$ is the distribution function of $Y$.