Expected value and variance of multivariate exponential distr.

[the_dane]

Homework Statement

https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png
I want to find variance matrix and expected variance vector of Y=(Y1,Y2). Y1 and Y2 are independent. Γ is the gamma function and ϒ is a known parameter. λ1>0 λ2>0 and ϒ>0

Homework Equations

canonical form formulas for exponential family.

The Attempt at a Solution

I want to do it by proving that it belongs to the exponential family and then bring to canonical form. But in our textbook (Dobson, Generalized linear model) there are only formulas for f(y) where y is not a vector as here y=(y1,y1)

 

[Ray Vickson]

Homework Statement

https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png
I want to find variance matrix and expected variance vector of Y=(Y1,Y2). Y1 and Y2 are independent. Γ is the gamma function and ϒ is a known parameter. λ1>0 λ2>0 and ϒ>0

Homework Equations

canonical form formulas for exponential family.

The Attempt at a Solution

I want to do it by proving that it belongs to the exponential family and then bring to canonical form. But in our textbook (Dobson, Generalized linear model) there are only formulas for f(y) where y is not a vector as here y=(y1,y1)

Are the $y_i$ discrete (integer-valued) or continuous? If continuous, what do you mean by $y_i!$, etc? If they are discrete, is your function $p$ a joint probability mass function? If they are continuous, is your function $p$ a joint probability density function? Why do you write $p(y_1+y_2)$ when the function on the right is not just a function of $y_1+y_2$, but is a function of $(y_1,y_2)$?

Why do you say that $Y_1,Y_2$ are independent? They do not look independent at all, judging by their joint distribution that you give.

Anyway, you are supposed to do some work on a problem before bringing it to this forum; we are not allowed to help you until you show what you have done already.

 

[the_dane]

They are discrete and I was wrong saying that they are independent.

I know I have to do some work before, but I brought it because I'm totally stuck. I think my problem is that in this course our textbook only have examples for one variable and not multi. That confuses me.

 

[Ray Vickson]

They are discrete and I was wrong saying that they are independent.

I know I have to do some work before, but I brought it because I'm totally stuck. I think my problem is that in this course our textbook only have examples for one variable and not multi. That confuses me.

Start by trying to obtain formulas for the marginal distributions of $Y_1$ and $Y_2$. You can get the means and variances of $Y_1$ and $Y_2$ from their marginals. That leaves only the problem of finding $\text{Cov}(Y_1,Y_2)$, which must use the full bivariate distribution $p(y_1,y_2)$.

BTW: you still have not specified the problem very well: do $y_1, y_2$ belong to the integers $\{0,1,2, \ldots \}$?