Conditional joint distribution

Definition:

• Conditional Distribution and Joint Distribution
• Takes $O(d^n)$ to store the joint distribution

Discrete case:

• $p_{X|Y}(x|y)=P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}=\frac{p(x,y)}{p_Y(y)}$
• for all $y$ such that $p_Y(y)>0$
  • $F_{X|Y}(x|y)=P(X\le x|Y\le y)=\sum _{a\le x}{p}_{X|Y}(a|y)$

Continuous case:

• $f_{X|Y}(x|y)=P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}=\frac{f(x,y)}{f_Y(y)}$
  • for all $y$ such that $f_Y(y)>0$
• If $X$ and $Y$ are jointly continuous then the Joint cumulative distribution function:
  • $P(X\in A|Y=y)={\int }_A{f}_{X|Y}(x|y)dx$
  • $F_{X|Y}(a|y)=P(X\le a|Y=y)={\int }_{-\infty }^a{f}_{X|Y}(x|y)dx$
    • contains $y$ after integration, otherwise independent

---