Expected value and variance of multivariate exponential distr.

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Homework Help Overview

The discussion revolves around finding the variance matrix and expected variance vector for a bivariate random variable Y=(Y1,Y2), where Y1 and Y2 are discrete random variables. The context involves the exponential family of distributions and the use of canonical forms, with references to gamma functions and parameters.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to prove that the random variable belongs to the exponential family and to derive the canonical form. Questions arise regarding the nature of the variables (discrete vs. continuous) and their independence.
  • One participant questions the independence of Y1 and Y2 based on their joint distribution, prompting a reevaluation of assumptions.
  • Another participant suggests starting with the marginal distributions to derive means and variances, while also addressing the covariance between Y1 and Y2.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the problem and questioning the assumptions made about the variables. Some guidance has been offered regarding the approach to finding marginal distributions and covariance, but no consensus has been reached on the specifics of the problem.

Contextual Notes

Participants note that the textbook used for the course primarily contains examples for single-variable cases, which contributes to confusion when dealing with multivariate scenarios. There is also a lack of clarity regarding the specification of the problem, particularly concerning the nature of the variables involved.

the_dane
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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)
 
Last edited by a moderator:
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the_dane said:

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.
 
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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.
 
the_dane said:
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 \}##?
 

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