# Expected Value of Nonnegative Identically Distributed Random Variables

Let $$X_1, X_2 \geq 0$$ be two non-negative identically distributed random variables. I wonder if the following equation holds.

$$\mathbb{E}[X_1X_2] = \mathbb{E}[Y^2]$$ where $$Y$$ is a random variable having the common distribution of $$\{X_1, X_2\}$$.

My attempt: We know that, in general, $$\mathbb{E}[X_1X_2] \neq \mathbb{E}[X_1^2]$$ and/or $$\mathbb{E}[X_1X_2] \neq \mathbb{E}[X_2^2]$$. One can take $$X_1 \in \{0,1\}$$ with probability $$1/2$$ and take $$X_2 := -X_1$$. However, the nonnegativity excludes this case.

Also, if we look at $$\mathbb{E}[Y^2] = \int y^2\, dF_{Y} = \int y^2 \, dF_{X_1,X_2}$$ where the last equality hold since $$Y$$ has the common joint distribution of $$\{X_1,X_2\}$$. But I get stuck to see if this is equal to $$\mathbb{E}[X_1X_2] = \int x_1x_2 \,dF_{X_1,X_2}$$. Any comment is appreciated.

• What if $X_1$ and $X_2$ are independent?
– user140541
Jul 26, 2021 at 8:35

It is easy to find such examples. If $$EX_1X_2=EX_1^{2} (=EX_2^{2})$$ then $$E(X_1-X_2)^{2}=EX_1^{2}+EX_2^{2}-2EX_1X_2=0$$ so $$X_1=X_2$$ a.s.. Can you come up with two non-negative r.v.'s with the same distribution (say $$exp(1)$$) which are not a.s. equal?