Conditional Expectation as a function of X

Question

I hope someone can clear the doubt on conditional distribution, in a text, it mentions:

To interpret the notation $f_{X \mid Y}(x \mid y)$, we need to be clear that there is nothing random about the random variable $Y$, since it is already fixed. What remains is a sub-population distribution in terms of $X$ (i.e. the reduced sample space of $\Omega_X$ where $Y$ happened). Consequently, the conditional PMF/PDF $f_{X \mid Y}(x \mid y)$ is a distribution in terms of $X$.

I have no trouble understanding this, since conditional just means looking into the reduced sample space of $X$ where $Y$ has happened for a specific set. But soon after, when conditional expectation is mentioned, it says:

There are two points to note here. First, the expectation of $\mathbb{E}[X \mid Y=y]$ is taken with respect to $f_{X \mid Y}(x \mid y)$. We assume that the random variable $Y$ is already fixed at the state $Y=y$. Thus, the only source of randomness is $X$. Secondly, since the expectation $\mathbb{E}[X \mid Y=y]$ has eliminated the randomness of $X$, the resulting function is in $y$.

A quick read online reveals that the expectation of $X$ given $Y$ can indeed be a function of $Y$ if we treat this conditional expectation as random variable. I am just slightly confused why the conditional expectation is not a function of $X$, since the underlying conditional distribution is a function of $X$.

Answer 1

Without getting very much into measure theory,consider the random vector $(X,Y)$ with density $f_{X,Y}(\cdot,\cdot)$ (wrt a dominating measure $\text d\mu(x,y)$) decomposed as $$f_{X,Y}(x,y)=f_Y(y)\times f_{X|Y}(x|y)$$ where

• $f_Y(\cdot)$ is a probability density (wrt the appropriate dominating measure $\text d\mu_2(y)$) attached with the random variable $Y$
• $f_{X|Y}(\cdot|y)$ is a probability density (wrt the appropriate dominating measure $\text d\mu_1(x)$) for (almost) every $y\in\mathcal Y$, attached with a random variable $Z_y$. In this notation, $y$ is a parameter of the density.

For a fixed value of $y\in\mathcal Y$, $f_{X|Y}(\cdot|y)$ can thus be understood as a regular density over the set $\mathcal X$ and $$f_{X|Y}(\cdot|y):\ x\longmapsto f_{X|Y}(x|y)$$ is a non-negative integrable (measurable) function on $\mathcal X$ such that $$\int_\mathcal Xf_{X|Y}(x|y)\text d\mu_1(x)=1$$ This means that, for a fixed value of $y\in\mathcal Y$, the expectation of $Z_y\sim f_{X|Y}(\cdot|y)$ can considered and, provided it exists for this specific value of $y\in\mathcal Y$, be defined as $$\mathbb E_y[X] = \int_\mathcal X xf_{X|Y}(x|y)\text d\mu_1(x)\tag{1}$$ It is usually written as $\mathbb E[X|Y=y]$. In the event (1) exists for all values of $y\in\mathcal Y$, the function $$\varphi:\ y \longmapsto \mathbb E[X|Y=y]$$ is rigorously defined. It can therefore be called to transform the random variable $Y$ into the new random variable $\varphi(Y)$. It is usually written as $\mathbb E[X|Y]$ and is equal to (1) when the realisation of $Y$ is equal to $y$.

As seen above, this random variable $\varphi(Y)=\mathbb E[X|Y]$ is not a function of the random variable $X$, even though they may be correlated with one another. Hence, the citation

First, the expectation of $\mathbb E[X∣Y=y]$ is taken with respect to $f_{X∣Y}(x∣y)$. We assume that the random variable $Y$ is already fixed at the state $Y=y$. Thus, the only source of randomness is $X$.

should be restated as [with highlighted changes]:

First, the expectation $\mathbb E[X∣Y=y]$ is taken with respect to the distribution with density $f_{X∣Y}(x∣y)$. We assume that the random variable $Y$ is already observed at the realisation $Y=y$. Thus, the only remaining source of randomness in the expectation is $X$ with distribution $f_{X∣Y}(\cdot∣y)$, that is, the conditional distribution of $X$ given $Y=y$.

Similarly,

Secondly, since the expectation $\mathbb E[X∣Y=y]$ has eliminated the randomness of $X$, the resulting function is in $y$.

should state

Secondly, since the expectation $\mathbb E[X∣Y=y]$ has eliminated the (remaining) conditional randomness of $X$ given $Y=y$, the resulting function is a function of $y$.

Answer 2

If you take the expectation of $X$, it's not a function of $X$. You integrate $xf(x)$ over $x$, and so $x$ is gone, producing only a number.

If you take the expectation of $X|Y=y$, it's not a function of $X$. You integrate $xf(x|y)$ over $x$ (think of it as $xf(x)$ with a different $f(x)$ for different $y$), and so $x$ is gone, producing only a number, which may be different for different $y$s, hence a function of $y$.