Multiple Random Variables

Joint Probability Distribution Function

\[\mathbf{P}(a\leq X\leq b,c\leq Y\leq d)=\int_{c}^{d}\int_{a}^{b}f_{X,Y}(x,y)\,\mathrm{d}x\,\mathrm{d}y\]

Marginal Probability Density Function

\[f_X(x)=\int f_{X,Y}(x,y)\,\mathrm{d}y\] \[f_Y(y)=\int f_{X,Y}(x,y)\,\mathrm{d}x\]

Joint Cumulative Distribution Function

\[F_{X,Y}(x,y)=\mathbf{P}(X\leq x,Y\leq y)=\int_{-\infty}^{y}\int_{-\infty}^{x}f_{X,Y}(s,t)\,\mathrm{d}s\,\mathrm{d}t\] \[f_{X,Y}(x,y)=\dfrac{\partial^2F_{X,Y}(x,y)}{\partial x\,\partial y}\]