Miscellaneous

Bias and Confounding

A bias is any systematic error that pushes your estimate away from the truth. That includes things like selection bias, information bias or misclassification, and confounding. So bias is the big category; confounding is one member of that category.

Confounding happens when the apparent association between an exposure and an outcome is distorted by a third variable that is related to both. Classic example: if you study coffee and heart disease without accounting for smoking, smoking can make coffee look more harmful than it really is because smokers may drink more coffee and also have higher heart disease risk. The distortion is not due to bad measurement or bad sampling; it is due to mixing of effects.

So the relationship is:

• Bias = umbrella term for systematic error
• Confounding = a specific kind of systematic distortion

Not all bias is confounding, and not all problems in a study come from confounding. A useful way to keep them straight: confounding is about comparability of groups, while other biases are often about who got into the study or how variables were measured.

Dice

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

Math

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

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	$\sqrt{3}$	$1$	$\tfrac{1}{\sqrt{3}}$	$0$