期望
离散型:
E(X)=∑k=1+∞xkpk
E(X)=\sum_{k=1}^{+\infty}x_kp_k
E(X)=k=1∑+∞xkpk
连续型:
E(X)=∫−∞+∞xf(x)dx
E(X)=\int_{-\infty}^{+\infty}xf(x)dx
E(X)=∫−∞+∞xf(x)dx
若Y=g(X)Y=g(X)Y=g(X),其中ggg是连续函数,则
如果XXX是连续型:
E(X)=E[g(X)]=∑k=1+∞g(xk)pk
E(X)=E[g(X)]=\sum_{k=1}^{+\infty}g(x_k)p_k
E(X)=E[g(X)]=k=1∑+∞g(xk)pk
如果XXX是连续型:
E(X)=E(g(X))=∫−∞+∞g(x)f(x)dx
E(X)=E(g(X))=\int_{-\infty}^{+\infty}g(x)f(x)dx
E(X)=E(g(X))=∫−∞+∞g(x)f(x)dx
其余几个性质:
- E(C)=CE(C)=CE(C)=C
- E(CX)=CE(X)E(CX)=CE(X)E(CX)=CE(X)
- E(X+Y)=E(X)+E(Y)E(X+Y)=E(X)+E(Y)E(X+Y)=E(X)+E(Y)
- XXX、YYY相互独立,则E(XY)=E(X)E(Y)E(XY)=E(X)E(Y)E(XY)=E(X)E(Y)
若Z=g(X,Y)Z=g(X,Y)Z=g(X,Y),其中ggg连续,而且XXX和YYY的概率密度是f(x,y)f(x,y)f(x,y),则有
E(Z)=E(g(X,Y))=∫−∞+∞∫−∞+∞g(x,y)f(x,y)dxdy
E(Z)=E(g(X,Y))=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}g(x,y)f(x,y)dxdy
E(Z)=E(g(X,Y))=∫−∞+∞∫−∞+∞g(x,y)f(x,y)dxdy
方差
描述数据离散程度的量。
D(X)=Var(X)=E{[X−E(X)]2}
D(X)=Var(X)=E\{[X-E(X)]^2\}
D(X)=Var(X)=E{[X−E(X)]2}
离散型:
D(X)=∑k=1+∞[xk−E(X)]pk
D(X)=\sum_{k=1}^{+\infty}[x_k-E(X)]p_k
D(X)=k=1∑+∞[xk−E(X)]pk
连续型
D(X)=∫−∞+∞[x−E(X)]2f(x)dx
D(X)=\int_{-\infty}^{+\infty}[x-E(X)]^2f(x)dx
D(X)=∫−∞+∞[x−E(X)]2f(x)dx
一般公式
D(X)=E(X2)−[E(X)]2
D(X)=E(X^2)-[E(X)]^2
D(X)=E(X2)−[E(X)]2
几个性质:
- D(CX)=C2D(X)D(CX)=C^2D(X)D(CX)=C2D(X)
- D(X+C)=D(X)D(X+C)=D(X)D(X+C)=D(X)
- D(X+Y)=D(X)+D(Y)+2E{[X−E(X)][Y−E(Y)]}D(X+Y)=D(X)+D(Y)+2E\{[X-E(X)][Y-E(Y)]\}D(X+Y)=D(X)+D(Y)+2E{[X−E(X)][Y−E(Y)]},若X、YX、YX、Y独立,则D(X+Y)=D(X)+D(Y)D(X+Y)=D(X)+D(Y)D(X+Y)=D(X)+D(Y)