Correlation between two variables

Question

Assume $X_1$, $X_2$, $X_3$,..., $X_n$ are i.i.d, say that $Y_1$ = $X_1^2/\sum_i X_i^2$ and $Y_2$ = $X_2^2/\sum_i X_i^2$, how to calculate the correlation between $Y_1$ and $Y_2$ or prove that they are negatively correlated? Thanks.

Answer 1

First of all, note that the expectation of both $Y_1$ and $Y_2$ is 1/n. As mentioned above by user2875124, the formula for correlation is given by

$E[Y_1*Y_2] - E[Y_1]*E[Y_2]$

In order to prove that there is a negative correlation, we want to show that $E[Y_1*Y_2] <1/n^2$

We can write

$Y_1*Y_2=(X_1^2/\sum{X_i^2})*(X_2^2/\sum{X_i^2})=(1-\sum_{i\neq1}(X_i^2))*(X_2/\sum_{i,j}(X_i^2))$

Thus we can write the expectaiton

$E[Y_1*Y_2]=1/n-((n-2)/n^2+E[X_2^4/\sum_{i,j}(X_i^2)])$

Now, by jensen's inequality (shoutout again to user2875124), we can conclude that $E[X_2^4/\sum_{i,j}(X_i^2)]>1/n^2$

Putting these pieces together, we have that

$E[Y_1*Y_2]<1/n-(n-2)/n^2-1/n^2=1/n^2$

Therefore $Y_1$ and $Y_2$ are negatively correlated.

Answer 2

Note that $Y_1$ and $Y_2$ are identically distributed.

We have

$\text{cov}(Y_1,Y_2) = E[Y_1Y_2] - E[Y_1]E[Y_2] = E[X_1^2X_2^2/(\sum_i X_i^2)^2]-E[X_1^2/\sum_i X_i^2]^2$

Can you use this? Can Jensen's inequality be applied?