توزیع احتمال توأم

${\displaystyle \ p\{X\leq y\leq b\}=\int _{-\infty }^{a}\int _{-\infty }^{b}f(x,y)dxdy}$

${\displaystyle \ f(a,b)={\tfrac {d^{2}}{dadb}}F(a,b)}$

۳.${\displaystyle \ f_{X}(x)=\int _{-\infty }^{\infty }}$

۴. ${\displaystyle \ p\{a<X<a+da,b<y<=b+db\}=\int _{b}^{b+db}\int _{a}^{a+da}f(x,y)dxdy~f(a,b)dadb}$

${\displaystyle \ p\{X\in A,y\in B\}=p\{X\in A\}p\{y\in B\}}$

${\displaystyle \ F(a,b)=F_{X}(a)F_{Y}(b)}$

${\displaystyle \ f(y,x)=f_{X}(x)f_{Y}(y)}$

${\displaystyle \ F_{X+Y}(a)=p\{X+Y<=a\}=\int _{-\infty }^{\infty }\int _{-\infty }^{a-Y}f_{X}(x)f_{Y}(y)dxdy}$

${\displaystyle \ f_{X+Y}(a)=\int _{-\infty }^{\infty }f_{X}(a-y)f_{Y}(y)dy}$

${\displaystyle \ p_{X|Y}(x|y)=p\{X=x|Y=y\}={\tfrac {p\{X=x,Y=y\}}{p\{Y=y\}}}}$

${\displaystyle \ f_{X|Y}(x|y)={\tfrac {f(x,y)}{f_{Y}(y)}}}$

${\displaystyle \ F_{X|Y}(a|y)=p\{X<=a|Y=y\}=\int _{-\infty }^{a}f_{X|Y}(x|y)dx}$