Z U(0,2π)
f(z) = 0.5/π [0,2π]
f(z) = 0 其它 z
f(z) 为Z的概率密度函数.
Z的期望E(Z) = π,Z的方差D(Z) = π^2/3.
E(X) = ∫(0,2π) sin z f(z) dz = 0.5/π ∫(0,2π) sin zdz = - 0.5/π (cos 2π - cos 0)
= 0
E(Y) = ∫(0,2π) sin (z+K) f(z) dz = 0.5/π ∫(0,2π) sin (z+K) d(z+K)
= - 0.5/π [cos (2π+K) - cos (K)] = 0
D(X) = 0.5/π ∫ (0,2π) sin^2 z dz = 0.5
D(y) = 0.5/π ∫ (0,2π) sin^2 (z+K) dz = 0.5 - 0.125/π sin 2K
ρxy = E[XY]/[D(x)D(y)]^0.5 = cos K D(x) / [D(x)D(y)]^0.5
= cos K [D(x)/D(y)]^0.5 = cos K /(1 - 0.625 sin K /π) (1)
讨论：
1,K = 0 时,ρxy = 1,此时X=Y,X与Y完全相关；
2,K = π/2时,ρxy = 0,此时X与Y相位差90度,X与Y正交,相关系数为0；
3,k = π时,ρxy = - 1,此时X = - Y,X,Y反向相关最大.
4,相关系数为0,只是X,Y不相关；不表示二者独立.反之,二者独立必不相关.