6.2 Conditional Expectations

Expectations for conditional random variables are defined in the obvious way. Conditional expectations are given by

1. $𝖤[X\mid Y=y]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}s{f}_{X\mid Y}(s\mid y)\mathrm{d}s$,

2. $𝖤[Y\mid X=x]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}t{f}_{Y\mid X}(t\mid x)\mathrm{d}t$.

$𝖤[Y\mid X=x]$ is a function $g(x)$, say, of $x$ (a real number). If we have not yet seen $x$ then this becomes a function $g(X)$ of the random variable $X$. i.e. $𝖤[Y\mid X]$ is a random variable because it is a function of the random variable $X$.

Sometimes conditioning provides an easy way to obtain the expectations of the marginal variables. Consider the random variable $𝖤[h(Y)|X]$, which is a function of $X$. Just as $𝖤\left[g(X)\right]=\int g(s)f_X(s)ds$, so the expectation of $𝖤[h(Y)|X]$ is

Now consider $𝖤[g(X)h(Y)|X]$, which is a random variable, since it is a function of the random variable $X$.

Intuitively, by conditioning on the unknown $X$ it becomes an unknown constant as far as the expectation is concerned and so it can be taken outside the expectation.

Solution. 

1. (a)

   $𝖤[Y|X=x]=\alpha x$ and $\mathrm{𝖵𝖺𝗋}[Y|X=x]=1$.

2. (b)

   $𝖤\left[X\right]=0$ and

   	$𝖤\left[Y\right]$	$=𝖤\left[𝖤[Y|X]\right]$
   		$=𝖤\left[\alpha X\right]=\alpha 𝖤\left[X\right]=0$

3. (c)

   	$𝖤\left[XY\right]$	$=𝖤\left[𝖤[XY|X]\right]$
   		$=𝖤\left[X𝖤[Y|X]\right]$
   		$=𝖤\left[\alpha X^2\right]$
   		$=\alpha .$

   Note that $𝖤\left[XY\right]-𝖤\left[X\right]𝖤\left[Y\right]=\alpha$.

The conditional variances are given by

If $X$ and $Y$ are independent the conditional distributions are the same as the marginal distributions ($f_{X|Y}(x|y)=f_X(x)$ and $f_{Y|X}(y|x)=f_Y(y)$), so that in particular

1. $𝖤[X\mid Y=y]=𝖤\left[X\right]$,

2. $\mathrm{𝖵𝖺𝗋}[X\mid Y=y]=\mathrm{𝖵𝖺𝗋}\left[X\right]$,

3. $𝖤[Y\mid X=x]=𝖤\left[Y\right]$,

4. $\mathrm{𝖵𝖺𝗋}[Y\mid X=x]=\mathrm{𝖵𝖺𝗋}\left[Y\right]$.