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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.

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