1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Jun 9, 2010 #1
    Not really a homework question, but here goes...

    1. The problem statement, all variables and given/known data

    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?

    2. Relevant 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)?

    3. 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.
     
    Last edited: Jun 9, 2010
  2. jcsd
  3. Jun 10, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    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
     
    Last edited: Jun 10, 2010
  4. Jun 10, 2010 #3

    lanedance

    User Avatar
    Homework Helper

    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.
     
  5. Jun 11, 2010 #4
    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.
     
  6. Jun 12, 2010 #5

    lanedance

    User Avatar
    Homework Helper

    now worries, let me know if you want me to have a look at it
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Find the pmf of Y, given the pmf of X and Corr(X,Y).
  1. Joint PMF of p x,y (Replies: 2)

Loading...