联合概率分布可以通过对另一个随机变量积分或求和得到边缘分布：
离散P(X=x)=∑yf(x,y),P(Y=y)=∑xf(x,y). P(X = x) = \sum_y f(x, y), \quad P(Y = y) = \sum_x f(x, y). P(X=x)=y∑f(x,y),P(Y=y)=x∑f(x,y).
连续fX(x)=∫−∞∞f(x,y) dy,fY(y)=∫−∞∞f(x,y) dx.f_X(x) = \int_{-\infty}^\infty f(x, y) \, dy, \quad f_Y(y) = \int_{-\infty}^\infty f(x, y) \, dx. fX(x)=∫−∞∞f(x,y)dy,fY(y)=∫−∞∞f(x,y)dx.
X,YX,YX,Y的协方差
Cov(X,Y)=E[XY]−E[X]E[Y]=∬R2xyf(x,y)dxdy−∫−∞∞xf(x)dx∫−∞∞yf(y)dyCov(X,Y)=E[XY]-E[X]E[Y] =\iint_{R^{2}}xyf(x,y)dxdy-\int_{-\infty}^{\infty}xf(x)dx\int_{-\infty}^{\infty}yf(y)dy Cov(X,Y)=E[XY]−E[X]E[Y]=∬R2xyf(x,y)dxdy−∫−∞∞xf(x)dx∫−∞∞yf(y)dy
Tip
X与Y独立↔f(x,y)=f(x)f(y)X与Y独立 \leftrightarrow f(x,y)=f(x)f(y) X与Y独立↔f(x,y)=f(x)f(y)