一道英文概率题 X and Y have a multivariate discrete distribution with the probability functionf(X,Y) where f(-1,1) = f(0,0) = f(1,1) = 1/3.What is the correlationcoefficient between X and Compute the marginal density,fx(1) of X when itequals 1.Similarly compute fy(1).Does f(1,1) = fx(1)fy(1) Use this exampleto outline the relationship between correlation and independence.
问题描述：
一道英文概率题
X and Y have a multivariate discrete distribution with the probability function
f(X,Y) where f(-1,1) = f(0,0) = f(1,1) = 1/3.What is the correlation
coefficient between X and Compute the marginal density,fx(1) of X when it
equals 1.Similarly compute fy(1).Does f(1,1) = fx(1)fy(1) Use this example
to outline the relationship between correlation and independence.
答
1.先算出 mean of X and mean of Y:m(X)=1/3(-1+0+1)=0; m(Y)=1/3(1+0+1)=2/3
2.计算Covariance(X,Y)=(1/3)(-1)(1-2/3)+(1/3)(0)(0-2/3)+(1/3)(1)(1-2/3)=0,所以推出correlation coefficient is 0
3.fx(1)=1/3; fy(1)=2/3; However,1/3=f(1,1) 不等于 fx(1)fy(1)=2/9 从而推出X,Y are NOT independent.
This example shows us that uncorrelation does not imply independence.