The Probabilistic Cauchy Schwartz and the Analysis Cauchy Schwartz.

I’d often seen two different versions of Cauchy Schwartz (CS).  In analysis and linear algebra I’d learned that if x,y are elements of a vector space V we have

(1)   \begin{align*} \vert \left<x,y\right>\vert &\leq \Vert x\Vert \cdot \Vert y\Vert \end{align*}

while in probability and statistics I’d learned that for random variables x,y we have

(2)   \begin{align*} \vert E(xy)\vert &\leq \sqrt{E(x^2)}\sqrt{E(y^2)} \end{align*}

I’d seen proofs of both but not in a way that tied them together.  Today, I learned how they’re related and it’s super simple.  Define \left<x,y\right>=E(xy).  One can verify that this satisfies the definition of an inner product.  Then we have

(3)   \begin{align*} \vert E(xy)\vert &=\vert \left<x,y\right>\vert\\ &\leq \Vert x\Vert \cdot \Vert y\Vert\textrm{ by Analysis CS}\\ &= \sqrt{\left<x,x\right>}\sqrt{\left<y,y\right>}\\ &=\sqrt{E(x^2)}\sqrt{E(y^2)} \end{align*}

it seems so obvious but I’d never realized this previously.  Further, the statement

(4)   \begin{align*} \vert Cov(x,y)\vert &\leq \sqrt{Var(x)}\sqrt{Var(y)} \end{align*}

becomes really easy to prove using the inner product defined above.

(5)   \begin{align*} \vert Cov(x,y)\vert &=\left<x-E(x),y-E(y)\right>\\ &\leq \sqrt{\left<x-E(x),x-E(x)\right>}\sqrt{\left<y-E(y),y-E(y)\right>}\textrm{ by CS}\\ &=\sqrt{Var(x)}\sqrt{Var(y)} \end{align*}

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