Bernoulli distribution

$P(X=1)=p$, $P(X=0)=1-p$. $\lb X\rb=p$, $\lb X^2\rb=p$, $\Var=p(1-p)=pq$.

Origin: Bernoulli trials

A random experiment that has exactly two possible outcomes: success $X=1$ with probability $0<p<1$ and failure $X=0$ with probability $q=1-p$.

Binomial distribution

$X\sim \mathcal{B}(N,p)$, $P(X=n)=\binom{N}{n}p^nq^{N-n}$. $\lb X\rb=Np$, $\lb X^2\rb=Np$, $\Var=np(1-p)=Npq$.

Origin: $n$ independent Bernoulli trials

Origin: 1D random walk

Origin: Independent two-state switching processes

Poisson distribution $X\sim \text{Pois}(λ)$

Geometric distribution

Exponential distribution

Normal distribution