Common Distributions
| 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.