David Bindel
@dbindel.bsky.social
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Professing computing and applied math at Cornell. Numerical methods for data science, plasma physics, other stuff depending on the day. Director, Cornell Center for Applied Mathematics; Director, Simons Collaboration on Hidden Symmetries and Fusion Energy.
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![A short proof: here is the LaTeX code.
**Proof.** We have, for any $\color{blue}{\lambda} \in\mathbb{R}$,
\begin{align*}
\mathbb{E}[(X-\color{blue}{\lambda})^2]
&= \mathbb{E}[(X-\color{red}{\mathbb{E}[X]} + \color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})^2] \\
&=\mathbb{E}[(X-\color{red}{\mathbb{E}[X]})^2 + 2(X-\color{red}{\mathbb{E}[X]})(\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda}) + (\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})^2]\\
&=\underbrace{\mathbb{E}[(X-\color{red}{\mathbb{E}[X]})^2]}_{=\textrm{Var}[X]} + 2\underbrace{\mathbb{E}[X-\color{red}{\mathbb{E}[X]}]}_{=0}(\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})] + (\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})^2
\end{align*}
and that's all. (The first step is a trick known as *"hiding zero:"* writing $0=a-a$. 🤷)](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ac6qioenkpl3jmmnib3tpbhb/bafkreieevv3petzx4kqxx3o6i4oqtugsjh52n7i2ellidxmetefboag6fi@jpeg)
Reposted by David Bindel
Reposted by David Bindel
Reposted by David Bindel
Reposted by David Bindel
