how to compute expectation of $R_1R_2$ when $R_1$ and $R_2$ are independent?

Question

Here's my problem:

Let $R_1$ and $R_2$ be independent random variables, each with the exponential density $f(x) = e^{-x}, x \ge 0; f(x) = 0, x \lt 0$ Then, find the expectation of $R_1R_2$

I know that if $R_1$ and $R_2$ are independent, their probability distribution function equals $f_{12}(x_1, x_2) = f_1(x_1)f_2(x_2)$ when $f_1(x_1)$ and $f_2(x_2)$ are the respective probability distribution functions. But is there a more direct way of computing $f_{12}(x_1, x_2)$ than that by more fundamental fact(or definition)? Or is this the definition of independency of random variables?

Answer 1

You can also use the following argument: $ER_1R_2=E(E(R_1R_2)|R_2))=E(R_2E(R_1|R_2))=ER_2(ER_1)=ER_2 (1)=(1)(1)=1$.