Question

设随机变量X与Y独立且存在期望和方差，证明： D(XY)≥D(X)D(Y)

Answer 1

【答案】：因为X与Y独立，当然X²与Y²也独立，于是：
D(XY)=E(X²Y²)-[E(XY)]²=E(X²)E(Y²)-(E(X))²(E(Y))²
D(X)D(Y)=[E(X²)-(E(X))²[E(Y²)-(E(Y))²]
=E(X²)E(Y²)-E(X²)(E(Y))²-(E(X))²E(Y²)+(E(X))²(E(Y))²
故D(XY)-D(X)D(Y)=E(X²)(E(Y))²+(E(X))²E(Y)²-2(E(X))²(E(Y))²
=[E(X²)-(E(X))²](E(Y))²+(E(X))²[E(Y²)-(E(Y))²]
=D(X)(E(Y))²+(E(X))²D(Y)≥0
即证得D(XY)≥D(X)D(Y)．这里用的是方差的计算式，千万勿将E(X²)与(E(X))²搞混!