Conditional pmf

Definition

Let

• $X_1,X_2$ : Discrete random variables, with
  • $S_{X_1}$ : Support of $X_1$
  • $S_{X_2}$ : Support of $X_2$
  • Joint pmf $p_{X_1},p_{X_2}(x_1,x_2)$, positive on support $\mathcal{S}$, zero elsewhere
• $p_{X_1}(x_1)$ : Marginal pmf of $X_1$
• $p_{X_2}(x_2)$ : Marginal pmf of $X_2$

Suppose $x_1\in S_{X_1}$; hence, $p_{X_1}(x_1)>0$

Then by conditional probability,

$$\begin{array}{rl}p(X_2=x_2|X_1=x_1)& =\frac{P(X_1=x_1,X_2=x_2)}{P(X_1=x_1)}\\ & =\frac{p_{X_1,X_2}(x_1,x_2)}{p_{X_1}(x_1)},\forall x_2\in S_{X_2}\end{array}$$

And

• We denote $$ p_{X_{2}|X_{1}}(x_{2}|x_{1}) = \frac{p_{X_{1},X_{2}}(x_{1},x_{2})}{p_{X_{1}}(x_{1})}, \quad x_{2}\in S_{X_{2}}

- We call $p_{X_{2}|X_{1}}(x_{2}|x_{2})$ the **conditional pmf** of $X_{2}$ given that $X_{1}=x_{1}$ ## Remark Notice that $\forall x_{1}\in S_{X_{1}}$, 1. $p_{X_{2}|X_{1}}(x_{2}|x_{1})$ is nonnegative 2. $$ \begin{align} \sum_{x_{2}}p_{X_{2}|X_{1}}(x_{2}|x_{1}) & = \sum_{x_{2}} \frac{p_{X_{1},X_{2}}(x_{1},x_{2})}{p_{X_{1}}(x_{1})} \\ & = \frac{1}{p_{X_{1}}(x_{1})} \sum_{x_{2}} p_{X_{1},X_{2}}(x_{1},x_{2}) \\ & = \frac{p_{X_{1}}(x_{1})}{p_{X_{1}}(x_{2})} \\ & = 1 \end{align}

Therefore, $p_{X_2|X_1}(x_2,x_1)$ is satisfies properties of pmfs