I'm stuck trying to show $E(XY) = E(X)E(Y)$ for $X, Y$ nonnegative bounded independent random variables on a probability space. The definition of independence is that $P(\{ X \in B\} \cap \{ Y \in C\}) = P(X \in B) P(Y \in C)$ for Borel sets $B$ and $C$. I'm not assuming $X$ or $Y$ have probability density functions so I cannot use them. Nor can I use conditional expectation.
[Math] Expectation of product of independent random variables
expectationprobabilityprobability distributionsrandom variables
Best Answer
If two random variables $X,Y$ have a joint distribution then they are independent if and only if the corresponding CDF's satisfy: $$F_{X,Y}(x,y)=F_X(x)F_Y(y)\tag1$$ Here $(1)$ is a necessary but also sufficient condition for:$$\mathsf P_{X,Y}=\mathsf P_X\times \mathsf P_Y$$where $\mathsf P_{X,Y}$ denotes the probability on $(\mathbb R^2,\mathcal B^2)$ induced by $(X,Y):\Omega\to\mathbb R$ and $\mathsf P_X,\mathsf P_Y$ denote the probabilities on $(\mathbb R,\mathcal B)$ induced by $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$.
Then under suitable conditions: $$\mathsf EXY=\int xydF_{X,Y}(x,y)=\int\int xydF_X(x)dF_Y(y)=\int xdF_X(x)\int ydF_Y(y)=\mathsf EX\mathsf EY$$