Show that $Cov(X,Y) \geq -23$

Question

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$

can someone give me some hints on how to show it?(not an answer)

i know that $Cov(X,Y) = E(XY) - E(X)E(Y)$ and that $Var(X) = E(X^2) - (E(X))^2$ and the same for $Var(Y)$ but i cant find any pattern here..

Answer 1

Hint: Look at the Cauchy-Schwarz Inequality.

A different hint: What is the variance of the sum?

Answer 2

For convenience, define $X^\prime\stackrel{\rm{}def}{=} X-\mathbb{E}{X}$, $Y^\prime\stackrel{\rm{}def}{=} Y-\mathbb{E}{Y}$. Then $\operatorname{Var}X^\prime=\mathbb{E}[{X^\prime}^2]=\operatorname{Var}Y^\prime=23$, and $$\operatorname{cov}(X,Y)=\operatorname{cov}(X^\prime,Y^\prime)=\mathbb{E}[X^\prime Y^\prime.]$$

Now, by Cauchy—Schwarz, $$ \lvert \mathbb{E}[X^\prime Y^\prime] \rvert \leq \sqrt{\mathbb{E}[X^\prime]}\sqrt{\mathbb{E}[X^\prime]} = \sqrt{\operatorname{Var}X^\prime}\sqrt{\operatorname{Var}Y^\prime}$$