Conditional exp on an event

Definitions §

Conditioning on an event §

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space. Given an event $A\in \mathcal{F}$ such that $\mathbb{P}(A)>0$, and a random variable $X$, we definte the conditional expectation:

$$\mathbb{E}(X|A):=\frac{\mathbb{E}(X{1}_A)}{\mathbb{P}(A)}$$

Let’s check if that makes sense, i.e. it’s the same as using the coditional probability. First check the case $X$ is a discrete variable with values $S_X\subset \mathbb{R}$, which we can represent as a sum of indicators:

$$X=\sum _{x\in S_X}x1(X=x)$$ $$\frac{\mathbb{E}(X{1}_A)}{\mathbb{P}(A)}=\frac{1}{\mathbb{P}(A)}\sum _{x\in S_X}x\mathbb{E}(1_{x\cap A})=\sum _{x\in S_X}x\frac{\mathbb{P}(X=x\cap A)}{\mathbb{P}(A)}=\sum _{x\in S_X}x\mathbb{P}(X=x|A)$$

Conditioning on a discrete random variable §

Since we can represent a discrete random variable as a sum of indicators, it’s natural to define

$$\mathbb{E}(X|Y):=\sum _{y\in S_Y}1(Y=y)\mathbb{E}(X|Y=y)=\sum _{y\in S_Y}1(Y=y)\frac{\mathbb{E}(X1(Y=y))}{\mathbb{P}(Y=y)}$$

This is again a discrete random variable, and since it’s a sum of indicators of measurable sets $\sigma (Y)$, it’s $\sigma (Y)$ measurable. If we fix $\hat{y}\in S_Y$, and compute

$$\mathbb{E}[\mathbb{E}(X|Y)1(Y=\hat{y})]=\mathbb{E}\left[\sum _{y\in S_Y}1(Y=y)1(Y=\hat{y})\mathbb{E}(X|Y=y)\right]=\mathbb{E}[1(Y=\hat{y})\mathbb{E}(X|Y=y)]$$

but the expected value of $X$ conditioned to the event $Y=y$ is a number, hence we can pull it out

$$\mathbb{E}[1(Y=\hat{y})]\mathbb{E}(X|Y=y)=\mathbb{P}(Y=\hat{y})\mathbb{E}(X|Y=y)=\mathbb{E}(X1(Y=\hat{y}))$$

since any measurable set in $\sigma (Y)$ can be represented as a sum of elements $y$, this generalizes to the identity:

$$\mathbb{E}(\mathbb{E}(X|Y)1_A)=\mathbb{E}(X{1}_A)\forall A\in \sigma (Y)$$