# Beginner questions about applying Cauchy-Schwarz inequality correctly to RVs

Background: These are super boring questions but I'm trying to learn about CS inequality for probability... any help would be greatly appreciated. Thank you.

Say $$x = (1,2)$$ and $$y = (3,4)$$ then Cauchy-Schwarz says that

$$|1(3) + 2(4)| \le \sqrt{1^2 + 2^2} \sqrt{3^2 + 4^2}$$

$$3+8 \le \sqrt{5(25)} = 5\sqrt{5} \approx 11.18$$

Question 1: is that correct?

Now let $$X$$ be a RV where $$P(X=1) = P(X=2) = 1/2$$

and let $$Y$$ be a RV where $$P(Y=3) = P(Y=4) = 1/2$$

$$X$$ and $$Y$$ are independent

Then Cauchy Schwarz says $$|E(XY) | \le \sqrt{E[X^2]E[Y^2]}$$

$$XY$$ takes on values $$3,4,6,8$$ with probability $$1/4$$. So

$$| 3(.25) + 4(.25) + 6(.25) + 8(.25) | \le \sqrt{1(.5) + 4(.5)} \sqrt{9(.5) + 16(.5)}$$

$$| 21/4 | \le \sqrt{(2.5)(12.5)}$$

$$5.25 \le 5.59$$

Question 2: is that correct?

Question 3: It's interesting to me that in the non probability case there are 2 terms in the LHS, but in the probability case there are 4 terms on the LHS, ie $$|1(3) + 2(4)|$$ vs $$| 3(.25) + 4(.25) + 6(.25) + 8(.25) |$$ and this seems somewhat unsettling to me since $$xy$$ can take the 2 extra values 4 and 6... Does that strike anyone else as weird? It seems like the new RV $$XY$$ is not equivalent to the operation of taking the dot product of $$x$$ and $$y$$ because new terms are introduced.