# Math Help - joint density question - math stats

1. ## joint density question - math stats

mathematical statistics
let f(x) be a density on R+ (so f(x) < 0 if x < 0). Let g(x,y) = f(x+y)/(x+y), x > 0, y> 0
a) show g is a density on R^2
b) assume that the expectation u and variance sigma^2 associated univariate density f exist and that mu^2 does not equal 2sigma^2. Show that X and Y are dependent.

I have part a done. As for part b I am terribly confused. I first took the question as meaning E[X] = mu and E[Y] = mu. Then I assume that X,Y were independent for a proof by contradiction. So, E[XY] = mu^2 and Var(X+y) = 2sigma^2 (since cov(X,Y) = 0 and also assuming Var(X) = sigma^2 = Var(Y)). Then from the question we would assume that E[XY] does not equal Var(X+Y) from which I got nowhere.

So I thought I would have to go back and play with the joint density g(x,y) and use the definition of independence for a joint density: g(x,y) = g_Y(y) g_X(x) except I have no clue on how to the integral for either marginal.

Any hints as to how to attack this problem would be greatly appreciated

2. Originally Posted by muskie
mathematical statistics
let f(x) be a density on R+ (so f(x) < 0 if x < 0). Let g(x,y) = f(x+y)/(x+y), x > 0, y> 0
a) show g is a density on R^2
b) assume that the expectation u and variance sigma^2 associated univariate density f exist and that mu^2 does not equal 2sigma^2. Show that X and Y are dependent.

I have part a done. As for part b I am terribly confused. I first took the question as meaning E[X] = mu and E[Y] = mu. Then I assume that X,Y were independent for a proof by contradiction. So, E[XY] = mu^2 and Var(X+y) = 2sigma^2 (since cov(X,Y) = 0 and also assuming Var(X) = sigma^2 = Var(Y)). Then from the question we would assume that E[XY] does not equal Var(X+Y) from which I got nowhere.

So I thought I would have to go back and play with the joint density g(x,y) and use the definition of independence for a joint density: g(x,y) = g_Y(y) g_X(x) except I have no clue on how to the integral for either marginal.

Any hints as to how to attack this problem would be greatly appreciated
I think you understood part b) incorrectly. The question should have been, more explicitly: "Let (X,Y) be a r.v. distributed with density $g(x,y)=\frac{f(x+y)}{x+y}$. Let us assume (on the other hand) that the distribution with density $f$ has mean $\mu$ and variance $\sigma^2$. Show that $X$ and $Y$ are dependent if $\mu^2\neq 2\sigma^2$".

Thus $E[X]\neq \mu$ a priori. But $E[X]=\int\int x g(x,y) dx dy$.

Further hints: prove that $E[X]=E[Y]=\frac{\mu}{2}$ and that $E[XY]=\frac{1}{6}(\sigma^2+\mu^2)$, and conclude from there.