Common Distributions

November 3, 2018 - 1 minute read - Category: Ml - Tags: Probability
Distribution Name Explanation PMF Mean Var
Bernoulii(p) Con flip $(0,1)\to(p,1-p)$ $p$ $p(1-p)$
Binominal(n, p) Sum of n Bernouliis $m(k)={N\choose k}p^k(1-p)^{n-k}$ $np$ $np(1-p)$
Geometric(p) The first time 1 appears in sequential Bernouliis $m(k)=p(1-p)^{k-1}$ $1/p$ $1/p(1-1/p)$
Poission(λ) How many 1s in one period for low p Bin $m(k)=\frac{\lambda^k}{k!}e^{-\lambda}$ $\lambda$ $\lambda$
Exponential(λ ) How many times till first 1 appears with low p Bin $f(x)=\lambda e^{-\lambda x}$ $1/\lambda$ $1/\lambda^2$

Remark: How to construct a successful observation of a Poisson R.V?

  • $n$ uniform samples of points in interval $[0,1]$
  • equally, segment the interval into $k$ (a large number) sub-intervals.
  • The number of points in each sub-interval is a Poisson R.V.
  • With an attempt to observe the distribution of this Poisson, we shall compare the results for all sub-intervals and plot them.