DSE 2010 q23 (Correlation Y=x^2).

[Sumit]

Firstly From the equation y=x^2 we can deduce that X and Y is Dependent...But why it is Uncorrelated ...Isn't it Non-Linearly correlated????

But If we assume that question is asking for Linear correlation then by formula of Linear correlation by karl pearson {rxy=Cov(x,y)/S.D of x*S.D of y) linear correlation btw x and y is equal to zero....bcoz Cov(x,y)=E(x,y)-E(x)E(y)=zero minus zero=zero.

So, The answer is X and Y are uncorrelated and dependent.

 Also, We know that two variable X and Y are independent If E(xy)=E(x)E(y) satisfy....Here in this question
E(xy)=E(x^3)=0, E(x)=0 & E(y)=2/3.
Therefore, E(xy)=E(x)E(y)=0 but we know that x and y are dependent...Is this means we can't use this formula If any one of the expectation is zero????

PS: I do know the property of correlation that If two variable are independent than linear correlation btw them  is zero that is they are uncorrelated but converse is not true..

[Amit Goyal]

Independence of Random Variables X and Y
implies
X and Y are uncorrelated i.e. E(XY)=E(X)E(Y)
but not vice versa.
Please read the definition of independence of two random variables carefully from the textbook.

[duck]

Hi Sumit.. :)

Correlation coefficient tells you the degree of linear relationship between two variables. And that you correctly pointed out that Cov(X,Y)=0 . Hence, Correlation=0. Therefore, uncorrelated.

However, for independence the claim is:
If X and Y are independent then, E(XY)= E(X)*E(Y).

[Sumit]

thanks Amit sir & Duck I got My mistake....