202405302242
Status: #idea
Tags: Important Inequalities
State: #nascient

Cauchy-Schwarz's Inequality

It states that given two vectors $\overrightarrow{u},\overrightarrow{v}$ their dot product will always be lesser or equal to the product of the magnitudes, so:

In probability it becomes:

This inequality is cool because it allows us to put a bound on the covariance of two random variables. Indeed just replace the $X$ and $Y$ by $X-{\mu }_X$ and $Y-{\mu }_Y$ and from this inequality you can show that there's a bound on the covariance.

Not only that, by doing nothing more than simple manipulation of the above expression you obtain that ${\rho }^2$ is less or equal to $1$.

You can prove it using the Proof for Cauchy-Schwarz's Inequality using Semi-Positive Definite Matrix
You can also prove it directly without resorting to Linear Algebra.

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