# Marginal variance with two random variables

• Oct 19th 2010, 07:08 AM
uberbandgeek6
Marginal variance with two random variables
If ten fair six-sided dice are rolled, suppose that X is the total number of even numbers shown and Y is the total number of fives shown.
(a) What is the joint p.m.f. of X and Y ?
(b) What is the marginal variance of X? Can you answer this question in one line, using an argument that does not involve any summation?

I found for (a) that f(x,y) = ((1/2)^x)*((1/6)^y)*((1/3)^(10-x-y))*10!/(x!y!(10-x-y)!).
But I really don't know how to approach (b). How can you tell the marginal variance of X without doing all of the summations?
• Oct 20th 2010, 12:00 AM
CaptainBlack
Quote:

Originally Posted by uberbandgeek6
If ten fair six-sided dice are rolled, suppose that X is the total number of even numbers shown and Y is the total number of fives shown.
(a) What is the joint p.m.f. of X and Y ?
(b) What is the marginal variance of X? Can you answer this question in one line, using an argument that does not involve any summation?

I found for (a) that f(x,y) = ((1/2)^x)*((1/6)^y)*((1/3)^(10-x-y))*10!/(x!y!(10-x-y)!).
But I really don't know how to approach (b). How can you tell the marginal variance of X without doing all of the summations?

I think I would write the joint distribution as:

$f_{XY}(x,y)=p(y|x)p(x)=b(y,(10-x),1/3)b(x,10,1/2)$

or at least said that this is a multinomial distribution (they expand to the same thing and apperars to be what you have above).

CB
• Oct 20th 2010, 05:43 AM
CaptainBlack
Quote:

Originally Posted by uberbandgeek6
If ten fair six-sided dice are rolled, suppose that X is the total number of even numbers shown and Y is the total number of fives shown.
(a) What is the joint p.m.f. of X and Y ?
(b) What is the marginal variance of X? Can you answer this question in one line, using an argument that does not involve any summation?

It will take more than a line to explain it, but it is faily ellementary:

$var(x)=\sum_x\sum_y (x-\overline{x})^2 p(y|x)p(x)=\sum_x (x-\overline{x})^2 p(x)=10(1/2)^2$

or if you like the marginal variance must be the variance of the binomial distribution of $X$ without any reference to $Y$.

CB