Covariance/ Correlation Calculation

[ewoeckel]

1. Let N and T be the number of users logged on and the time until the next log-off. The joint probability of N and T is given by P(N=η, X≤t) = (1-ρ)ρ^{η-1}(1-e^{-ηλt}) for η=1,2,...;t>0.) Find the correlation and covariance of N and T.

2. COV(X,Y) = E[XY]-E[X]E[Y]
ρ_{X,Y} = (E[XY]-E[X]E[Y])/(σ_{X}σ_{Y})
E[XY] = ∫∫ xy f_{XY}(x,y) dx dy

3. I do not know how to find the expected value from the joint PDF and was hoping for some guidance.

 

[jrlaguna]

Same as with a single variable PDF. Assume f(x,y) is the normalized PDF

E[x]=\int x f(x,y) dx dy, etc.

But remember that "your PDF" is *not* a true PDF, it is integrated for t! (X<=t)!