Product of correlated random variables

[benjaminmar8]

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)?

 

[mXSCNT]

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.