Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Product of correlated random variables

  1. Apr 23, 2009 #1
    Hi, All,

    Let x1 x2... Xn be correlated random events (or variables). Say P(X1), P(X2)..., P(Xn) can be computed, in addition to that, covariance and correlated between all X can be computed. My question is, what is P(X1) * P(X2) *... * P(Xn)?
  2. jcsd
  3. Apr 23, 2009 #2
    Well, if P(X1=x1)...P(Xn=xn) can be computed, then obviously P(X1=x1) * ... * P(Xn=xn) can also be computed as the product of those. Perhaps you meant to ask, "what is P(X1=x1, X2=x2, ..., Xn=xn)?" From just the correlation coefficients, you don't have enough information to compute that. I could give you a specific example of a situation where P(X1=x1,X2=x2) can't be computed from your given information, but it is more intuitive to note that the correlation coefficients give you n^2 numbers, and if you know every P(X_i=x_j) that gives you mn numbers where m is the number of discrete values each variable may take, but knowing every P(X1=x_j1,...,Xn=x_jn) for each sequence of indices j1 ... jn, involves knowing m^n numbers, potentially a much larger number than n^2 + mn. So the full joint distribution usually has a lot more "degrees of freedom" than the correlation matrix + the individual distributions, so specifying the latter can't tell you everything about the former. That's not exactly a proof, but it should help give an intuitive idea.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook