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
- $\textbf{MGF}(t) = \textbf{E}_x (e^{tx}) = \int e^{tx} \text{PDF}(x) dx$
Inequalities
- $P( X > \epsilon) < E(X) / \epsilon$ , 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
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