Lecture 6. C) Conditional Moments

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Some Conditional Moments

  • The expected value of [math]X[/math] given [math]Y[/math] is defined as

[math]E_{\left.X\right|Y}\left(\left.X\right|Y=y\right)=\begin{cases} \sum_{s\in\mathbb{R}}s.f_{X|Y}\left(s,y\right), & \text{if}\,\left(X,Y\right)'\,\text{is discrete }\\ \int_{-\infty}^{\infty}s.f_{X|Y}\left(s,y\right)ds & \text{if}\,\left(X,Y\right)'\,\text{is continuous} \end{cases}[/math]

We often write [math]E_{\left.X\right|Y}\left(\left.X\right|Y\right)[/math] instead of [math]E_{\left.X\right|Y}\left(\left.X\right|Y=y\right)[/math], when we haven't yet settled for a value of [math]y[/math].

Perhaps a more popular notation, which means the same, is

[math]E\left(\left.X\right|Y\right).[/math]

Notice that [math]E\left(\left.X\right|Y\right)[/math] is a function of [math]Y.[/math] The [math]X[/math] has been integrated out!

  • The conditional variance of [math]X[/math] given [math]Y[/math] is

[math]Var\left(\left.X\right|Y\right)=E\left[\left.\left(X-E\left(\left.X\right|Y\right)\right)^{2}\right|Y\right]=E\left(\left.X^{2}\right|Y\right)-E\left(\left.X\right|Y\right)^{2}[/math].