Proof Tactics

Foundations §

• Mathematical Induction
• By Contradiction
• By Contra-positive: if a then b $⟺$ if not b then not a
• Falsity, by counter-example
  • specifically, you can prove a universal statement false, or a existential statement true

a If and only if b is proven by two legs:

1. Forward, if a then b
2. Backward, if b then a either of these legs can be then proven by above induction, contradiction, by contrapositive, or by any combination.

Algebraic

• You can split summations (even infinite ones)
• ${\prod }_i{e}^{X_i}=e^{{\sum }_i{X}_i}$
• Equivalently, $\ln \left({\prod }_i{X}_i\right)={\sum }_i(\ln X_i)$ Pushing log thru product makes it a sum
• Telescoping sums and products
  • Useful when moveing
• Break up summations ${\sum }_{\forall i}={\sum }_{i\le k}+{\sum }_{i>k}$
• Summation of Squared ${\sum }_{{\forall }_i}(a_i+b_i{)}^2=\sum a^2+\sum b^2+\underset{{\sum }_{i<j}2a_i{b}_i}{\underbrace{\sum _{i\ne j}{a}_i{b}_i}}$ Probability
• Tower property $\mathbb{E}(X)=\mathbb{E}(\mathbb{E}(X|\mathcal{F}))$
• Linearity of expectation. You can switch expectation and summation
• Definition of expectation is an integral (or summation)
• Two RVs are identical if they have the same CDF.¹

Calculus

• ${\int }_x{\int }_{y}dydx={\int }_y{\int }_{x}dxdy$ switch integrals

Inequalities

• Sum greater than max
• Max greater than averages
• Average greater than min. Combined:

$$\underset{\text{ sum }}{\underbrace{\sum _{i=1}^n{x}_i}}\ge \text{max}(x_1,\dots ,x_n)\ge \underset{\text{ average }}{\underbrace{\frac{1}{n}\sum _{i=1}^n{x}_i}}\ge \text{min}(x_1,\dots ,x_n)$$

$\ln (1-x)\le -x$ always.

Footnotes §

1. Why CDF? statistics - Why is there a preference to use the cumulative distribution function to characterise a random variable instead of the probability density function? - Mathematics Stack Exchange ↩