Conditional expectation

Given an event

Let $A$ be an event with $P(A)>0$. Given a discrete Random variable $Y$, the conditional expectation $Y$, given $A$, is expressed as

$$\mathbb{E}(Y\mid A)=\sum _{y\in \text{Support}(Y)}y\cdot P(Y=y\mid A)$$

If $Y$ is continuous, the expectation becomes an integral instead.

$$\mathbb{E}(Y\mid A)={\int }_{-\infty }^{\infty }y\cdot f_Y(y\mid A)dy$$

Here, $f_Y(y\mid A)$ is called the conditional PDF and is defined as the derivative of the conditional CDF $F_Y(y\mid A)=P(Y\le y\mid A)$.

Given a random variable

Let $X$ and $Y$ be random variables. Let $g$ be a function that takes any real number $x\in \text{Support(X)}$ and produces $g(x)=\mathbb{E}(Y\mid X=x)$. Then the conditional expectation of $Y$, given $X$, is given by the random variable defined by $g(X)$.

$$\mathbb{E}(Y\mid X)=g(X)$$

The distinction between what is a constant and what is a random variable is important.

• $\mathbb{E}(Y)$ remains a constant.
• $\mathbb{E}(Y\mid X=x)$ is also a constant.
• $\mathbb{E}(Y\mid X)$ is a random variable. It is defined as $g(X)$, which is a transformation of $X$, a random variable (here, we don’t know that $X=x$ for some $x\in \text{Support}(X)$).

Properties

• If $X$ and $Y$ are Independent, then $\mathbb{E}(Y\mid X)=\mathbb{E}(Y)$.
• $\mathbb{E}(h(X)\cdot Y\mid X)=h(X)\cdot \mathbb{E}(Y\mid X)$ for any function $h$. Intuitively, we are given $X$, so we can treat $h(X)$ as a constant and move it out under expectation properties. In other words, $c=h(X)$ and $\mathbb{E}(cY\mid X)=c\cdot \mathbb{E}(Y\mid X)$.
• $\mathbb{E}(Y_1+Y_2\mid X)=\mathbb{E}(Y_1\mid X)+\mathbb{E}(Y_2\mid X)$.