random variables with $\sigma(A,B)=1$ and gamma distributed

Question

We've discussed about the following (without solution) in class:

Let $(X_k)_{k=1}^\infty$ be a sequence of indipendent and uniformly distributed random variables in $[0,1]$.

Are there random Variables $A$ and $B$ such that we get the covariance $\sigma(A,B)=1$ and $A,B$ are distributed with $f(x)=x\exp(-x) \mathbf{1}_{x>0}$?

I want to use $(x_k)_{k=1}^\infty$ to build $A$ and $B$.

I thought about using the exponential distribution and indipendence. But unfortunately none of our examples worked.

So are there $A$ and $B$?

Answer 1

Here is one solution. Given a joint distribution $\varphi(a,b)$ of two random variables $A$ and $B$, find a map $\theta:[0,1]^2 \to \mathbb R^2$ such that $$ \int_0^1 \int_0^1 f(x,y) \, dx\, dy = \int_{\mathbb R} \int_{\mathbb R} f(\theta(a,b)) \varphi(a,b) \, da \, db .$$ You can use some abstract argument to show that such a map exists. And in fact, I have such an argument in a paper I wrote: http://www.math.missouri.edu/~stephen/preprints/concrete.html Theorem 2.1.

Now let $(A,B) = \theta(X_1,X_2)$. And look - you can throw most of the random variables away!

Answer 2

Let $A = X+Y$ and $B=Y+Z$, where $X,Y,Z$ are independent, mean 0 exponential RV's. But what does it have to do with the sequence $X_k$?