i*******a
发帖数: 376 | 1 http://en.wikipedia.org/wiki/Uncorrelated
If X and Y are independent, then they are uncorrelated. However, not all
uncorrelated variables are independent. For example, if X is a continuous
random variable uniformly distributed on [−1, 1] and Y = X^2, then X
and Y are uncorrelated even though X determines Y and a particular value of
Y can be produced by only one or two values of X.
如何证明上面例子里x和y是uncorrelated. 如果x＝[0,1], y=x^2，x and y还是
uncorrelated的吗？
多谢 | a********l
发帖数: 55 | 2 1. X and Y are uncorrelated because Cov(X,Y)=0.
2. If X is uniformly distributed on [0,1], then Cov(X,Y)>0 and hence X and Y
are not uncorrelated. | i*******a
发帖数: 376 | 3 多谢，不过我想问得是：为什么说 Cov(X,Y)=0？
也就是写出步骤证明为什么E(xy)=E(x)*E(y)? 同理，在第二个问题，为什么E(xy)>E(x
)*E(y)? 假设第一种情况下，X未必是uniformly分布的，只是symmetricly分布.
我对概率是白痴，这个简单的证明我想不明白，多谢
Y
【在 a********l 的大作中提到】
: 1. X and Y are uncorrelated because Cov(X,Y)=0.
: 2. If X is uniformly distributed on [0,1], then Cov(X,Y)>0 and hence X and Y
: are not uncorrelated.
| a********l
发帖数: 55 | 4 Just use the integral definition of expected value to evaluate E(XY), E(X)
and E(Y). (I suppose you know how to calculate definite integral of
polynomial.)
P.S. For non-uniform but symmetric distribution, I don't know. |
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