Miscellaneous

We prefer low variance over high bias. Bias means overfitting and we want to train our little model to succeed in delivering Shareholder Value™ in a "fast-paced dynamic environment".

$$\text{Mean-Squared Error} = MSE = \text{Variance} + \text{Bias}^2 \text{MSE} = Var[X] + (E[X] - \mu)^2$$

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For dice rolls, $E[X] = 3.5$ and $Var(E) = \frac{35}{12}$

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• $\sum_1^n{k} = \frac{n(n+1)}{2}$
• $\sum_1^n{k^2} = \frac{n(n+1)(2n+1)}{6}$
• $\sum_{k=0}^{\infin}{p^k} = \frac{1}{1 - p}$, $\forall -1 \leq p \leq 1$
• $\frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + f'(x)g(x)$
• $\int{\frac{1}{x}} = ln(x)$
• $\int{\frac{1}{1 + x^2}} = tan^{-1}(x)$

Cox's Theorem (#)

This is a foundational result in Probability and the Philosophy of Probability. Professor Cox showed that if you want a mathematical system of reasoning or inference (use numbers) about uncertainty (absence of predictability) that behaves in a complete (takes into account all data) consistent (many ways of reasoning leading to the same answer) way, then that system is the Theory of Probability.

See this video for a nice quick explanation:

Trig Table from Middle School

$\theta$	$0^\circ \, (0)$	$30^\circ \, \left(\tfrac{\pi}{6}\right)$	$45^\circ \, \left(\tfrac{\pi}{4}\right)$	$60^\circ \, \left(\tfrac{\pi}{3}\right)$	$90^\circ \, \left(\tfrac{\pi}{2}\right)$
$\sin\theta$	$0$	$\tfrac{1}{2}$	$\tfrac{1}{\sqrt{2}}$	$\tfrac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$1$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{\sqrt{2}}$	$\tfrac{1}{2}$	$0$
$\tan\theta$	$0$	$\tfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	Not defined
$\csc\theta$	Not defined	$2$	$\sqrt{2}$	$\tfrac{2}{\sqrt{3}}$	$1$
$\sec\theta$	$1$	$\tfrac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	Not defined
$\cot\theta$	Not defined	$\sqrt{3}$	$1$	$\tfrac{1}{\sqrt{3}}$	$0$