Question

一道英文概率题 看不懂请教

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 Y? 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.

Answer 1

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.

Answer 2

那个，我不是很知道如果要用英文写过程的话要怎么表达……

直接给个式子和答案吧：

1-0.25-0.35-0.10=0.30

其实这题从数学上说很简单，所以如果非要说过程的话……我想可不可以这样表达

X is a discrete random variable and It's possible values are 2, 4, 6, 8. so P(X=2) P(X=4) P(X=6) P(X=8)=1
now we know P(X=2)=0.25, P(X=4)=0.35, P(X=6)=0.10,
so P(X=8)=1-[P(X=2) P(X=4) P(X=6)]=1-0.25-0.35-0.10=0.30