Covariance between squared bivariate geometrically distributed random variables

Question

If $X$ and $Y$ follows bivariate geometric distribution (where $EX=a$, $EY=b$, $Cov(X,Y)=c$ ) then how to obtain (determine) $Cov(X^2,Y)$?

Answer 1

The particular bivariate Geometric you provide in your comment above is the model provided by:

• Phatak and Sreehari (1981), Some characterizations of bivariate geometric distribution, Journal of Indian Statistical Association, 19, pp. 141-146.

... namely with joint pmf $f(x,y)$:

_{(source: tri.org.au)}

Then, $Cov(X,Y)$ is:

_{(source: tri.org.au)}

and $Cov(X^2,Y)$ is:

_{(source: tri.org.au)}

where I am using the Cov function from the mathStatica package for Mathematica to help automate the calculation. As disclosure, I should add that I am one of the authors.