Find the pmf of Y, given the pmf of X and Corr(X,Y).

[jimholt]

Not really a homework question, but here goes...

Homework Statement

If we have two (binary) random variables X and Y, and we know the probability mass function for X, as well as the correlation between X and Y, can we find the probability mass function for Y?

Homework Equations

Let f(x) be the mass function for X and g(y) be the mass function for Y. Let g(1) = q and g(0) = 1-q. Given f(1) = p and f(0) = 1-p, and that the correlation between X and Y is c, can we find q as a function of p and c (only)?

The Attempt at a Solution

Let h(x,y) be the joint distribution of X and Y. We can show that that Cov(X,Y) = h(1,1)-pq, and so Corr(X,Y) = [h(1,1)-pq]/sqrt[pq(1-p)(1-q)] = c.

Where to go from there? Seems as if we have one equation with two unknowns. (Is there another equation lurking somewhere?)

Thanks for any help folks.

 

[lanedance]

I think what you've shown is that there is not necessarily a unique solution.

To confirm this, i would pick some random values and see if you can find multiple solutions

 

[lanedance]

As another example consider when X & Y are independent then h(x,y) = p(x)p(y).

The co-variance is Cov(X,Y) = h(1,1)-pq = pq-pq = 0, as expected.

Then q can be any value from 0 to 1.

 

[jimholt]

Yup, quite right. This brings up another confusing point for me, but let me chew on it a bit before starting a new thread. Thanks for lending me your eyes.

 

[lanedance]

now worries, let me know if you want me to have a look at it