Joint/Bivariate Probability Distributions

Joint PDF: $p(y_1, y_2)=P(Y_1=y_1, Y_2=y_2) = f(y_1, y_2)$

• $\forall y_1, y_2 \in \R, p(y_1, y_2) \ge 0$
• Discrete: $\sum_{y_1,y_2}p(y_1,y_2) =1$
• Continuous: $\int_{-\infty}^\infty\int_{-\infty}^\infty f(y_1, y_2) \, dy_1 \, dy_2 = 1$
• If $y_1,y_2$ are independent $\Leftrightarrow f(y_1,y_2)=f_1(y_1) \cdot f_2(y_2)$

Joint CDF: $F(y_1,y_2)=P(Y_1\le y_1,Y_2\le y_2)$

• $F(y_1,y_2)=\int_{-\infty}^{y_1}\int_{-\infty}^{y_2} f(t_1, t_2) \, dt_2 \, dt_1$

Marginal Probability

Marginal PDF: Probability with respect to one random variable

• Discrete: $p_1(y_1)=\sum_{y_2} p(y_1, y_2)$
• Continuous: $f_1(y_1)=\int_{-\infty}^\infty f(y_1, y_2) \, dy_2$

Conditional PDF:

• Discrete: $p(y_1|y_2)=P(Y_1=y_1|Y_2=y_2)=\frac{p(y_1, y_2)}{p_2(y_2)}$
• Continuous: $f(y_1|y_2)=\frac{f(y_1,y_2)}{f_2(y_2)}$

Conditional CDF:

• Discrete: $F(y_1|y_2)=P(Y_1\le y_1|Y_2 = y_2)$