maths / probability

• Moment-generating-function (MGF) and Fourier transform
  • https://stats.stackexchange.com/questions/238776/how-would-you-explain-moment-generating-functionmgf-in-laymans-terms/238937#238937
  • $\text{MGF}(t)={\text{E}}_x(e^{tx})=\int e^{tx}\text{PDF}(x)dx$
• Inequalities
  • $P(X>ϵ)<E(X)/ϵ$, X > 0
  • Chebyshev
    • $P(|X-EX|>k)<Var(X)/k^2$
    • $P(|X-EX|>K\sigma )<1/k^2$
  • Chernoff
    • idea: find a monotonic function to convert $P(X>a)=P(e^{tX}>e^{ta}$, then resort to Markov/MGF
    • see https://en.wikipedia.org/wiki/Chernoff_bound
• maths/probability/distribution
  • Bernoulli → Binomial
    • Bernoulli Event is a single event with probability of $p$. Bernoulli trial is executing a series of Bernoulli event, i.i.d.
    • Moments: $np$, $np(1-p)$
  • maths/probability/distribution/poisson
    • Binomial PDF approaches to Poisson as $n->\inf$ while keeping $np=\lambda$. See PoissonLimitTheorem
      • Understanding: there are infinite number of Bernoulli events happening at the same time all the time, but with super small probability. As a result, the average number of events happen within any fixed time interval is the same.
      • Binomial is the sum of success of events; Poisson is the sum of occurrence in a time period - here the time period time unit is the same as $\lambda$
      • To derive moments of Poisson, apply "$np=\lambda$" to moments of Binomial