A Probability Distributions

Here are the basic facts about the probability distributions we will need in these lecture notes. For a much longer list of important distributions, check this wikipedia page.

A.1 Discrete distributions:

Note: \((q=1-p)\)

	Parameters	Notation	Support	pmf	\({\mathbb{E}}[X]\)	\(\operatorname{Var}[X]\)
Bernoulli	\(p\in (0,1)\)	\(B(p)\)	\(\{0,1\}\)	\((q,p,0,0,\dots)\)	\(p\)	\(pq\)
Binomial	\(n\in{\mathbb{N}}, p\in (0,1)\)	\(b(n,p)\)	\(\{0,1,\dots, n\}\)	\(\binom{n}{k} p^k q^{n-k}\)	\(np\)	\(npq\)
Geometric	\(p\in (0,1)\)	\(g(p)\)	\(\{0,1,\dots\}\)	\(p q^k\)	\(q/p\)	\(q/p^2\)
Poisson	\(\lambda\in(0,\infty)\)	\(P(\lambda)\)	\(\{0,1,\dots\}\)	\(e^{-\lambda} \tfrac{\lambda^k}{k!}\)	\(\lambda\)	\(\lambda\)

A.2 Continuous distributions:

Note: the pdf is given by the formula in the table only on its support. It is equal to \(0\) outside of it.

	Parameters	Notation	Support	pdf	\({\mathbb{E}}[X]\)	\(\operatorname{Var}[X]\)
Uniform	\(a\lt b\)	\(U(a,b)\)	\((a,b)\)	\(\frac{1}{b-a}\)	\(\frac{a+b}{2}\)	\(\frac{(b-a)^2}{12}\)
Normal	\(\mu\in{\mathbb R},\sigma \gt 0\)	\(N(\mu,\sigma)\)	\({\mathbb R}\)	\(\frac{1}{\sigma \sqrt{2\pi}} e^{-\tfrac{(x-\mu)^2}{2 \sigma^2}}\)	\(\mu\)	\(\sigma^2\)
Exponential	\(\lambda\gt 0\)	\(\operatorname{Exp}(\lambda)\)	\((0,\infty)\)	\(\lambda e^{-\lambda x}\)	\(\tfrac{1}{\lambda}\)	\(\frac{1}{\lambda^2}\)