2.3 Conditional Distributions and Expectations

<< 2.2 Transformations: Bivariate Random Variables |

Definition: 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 $X_{1}=x_{1}$ ## Definition: Conditional pdf