Conditional Distribution

def. Conditional Distribution. let $X,Y$ be jointly distributed. Then the conditional probability distribution of $Y$ given a specific value of $X$ is the conditional probability distribution.

1. $X,Y$ are discrete:

$${\text{pmf}}_{Y|X=x}(y):=\mathbb{P}(Y=y|X=x)=\frac{\mathbb{P}(Y=y\wedge X=x)}{\mathbb{P}(X=x)}$$

2. $X,Y$ are continuous:

$$f_{Y|X=x}=\frac{f_{X,Y}(x,y)}{f_X(x)}$$

• if $X\perp Y$:

$$f_{Y|X=x}(y)=f_Y(y)$$

Visually:

…the red line height divided by the orange line is the p.d.f conditional distribution $f_{X|Y=y}(x)$. …the orange line is the value of the marginal distribution $f_Y$ at $Y=y$.

thm. Continuous Multiplication Rule. let $X,Y$ be jointly distributed. Then:

$$f_{X,Y}(x,y)=f_{X|Y=y}(x)\cdot f_Y(y)$$

thm. Rule of Average Conditional Probability. let $X,Y$, Then:

$$\begin{gathered} f_X={\int }_{-\infty }^{\infty }{f}_{X|Y=y}\cdot f_{Y}dy \\ \mathbb{P}(X=x)=\int \mathbb{P}(X=x|Y=y)\cdot f_Y(y)dy \end{gathered}$$