\[ \renewcommand{\P}{\mathop{\mathbb{P}}\nolimits} \newcommand{\E}{\mathop{\mathbb{E}}\nolimits} \newcommand{\var}{\mathop{\rm Var}\nolimits} \newcommand{\VaR}{\mathop{\rm VaR}\nolimits} \newcommand{\cte}{\mathop{\rm CTE}\nolimits} \newcommand{\cov}{\mathop{\rm Cov}\nolimits} \newcommand{\limsup}{\mathop{\rm limsup}} \newcommand{\liminf}{\mathop{\rm liminf}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \renewcommand{\d}{\, \mathrm{d}} \newcommand{\dP}{\, \mathrm{d}\mathbb{P}} \newcommand{\eps}{\varepsilon} \renewcommand{\emptyset}{\varnothing} \]
Appendix B — A list of well known probability distributions
This is a list of probabilty distributions and some of their basic properties that we will be using throughout the course. This list (all information below) will also be attached to your exam paper.
| Name | Notation | Parameters | Pmf | Mean | Variance |
|---|---|---|---|---|---|
| Bernoulli | \(\text{Bernoulli}(p)\) | \(p \in (0,1)\) | \(f(0)=1-p, \ f(1)=p\) | \(p\) | \(p(1-p)\) |
| Binomial | \(\text{Binomial}(n,p)\) | \(n \in \{1,2,\ldots\}\), \(p \in (0,1)\) | \(f(k)=\binom{n}{k} p^k (1-p)^{n-k}\) for \(k=0,1,\ldots,n\) | \(np\) | \(np(1-p)\) |
| Geometric | \(\text{Geometric}(p)\) | \(p \in (0,1)\) | \(f(k)=(1-p)^{k-1}p\) for \(k=1,2,\ldots\) | \(1/p\) | \((1-p)/p^2\) |
| Poisson | \(\text{Poisson}(\lambda)\) | \(\lambda>0\) | \(f(k)=\frac{\lambda^k}{k!} e^{-\lambda}\) for \(k=0,1,\ldots\) | \(\lambda\) | \(\lambda\) |
| Name | Notation | Parameters | Mean | Variance | |
|---|---|---|---|---|---|
| Beta | \(\text{Beta}(\alpha,\beta)\) | \(\alpha, \beta>0\) | \(f(x)=x^{\alpha-1} (1-x)^{\beta-1}/B(\alpha,\beta)\) for \(x \in (0,1)\) | \(\alpha/(\alpha+\beta)\) | \(\alpha\beta/((\alpha+\beta)^2 (\alpha+\beta+1))\) |
| Exponential | \(\text{Exp}(\lambda)\) | \(\lambda>0\) | \(f(x)=\lambda e^{-\lambda x}\) for \(x >0\) | \(1/\lambda\) | \(1/\lambda^2\) |
| Gamma | \(\text{Gamma}(\alpha,\beta)\) | \(\alpha,\beta>0\) | \(f(x)=\beta^\alpha x^{\alpha-1} e^{-\beta x}/\Gamma(\alpha)\) for \(x >0\) | \(\alpha/\beta\) | \(\alpha/\beta^2\) |
| Normal | \(\mathcal{N}(\mu,\sigma^2)\) | \(\mu \in \R\), \(\sigma^2>0\) | \(f(x)=\exp \left( -(x-\mu)^2/(2\sigma^2) \right)/\sqrt{2\pi\sigma^2}\) for \(x \in \R\) | \(\mu\) | \(\sigma^2\) |
| Uniform | \(\text{Unif}(a,b)\) | \(a,b \in \R\), \(a<b\) | \(f(x)=1/(b-a)\) for \(x \in (a,b)\) | \((a+b)/2\) | \((b-a)^2/12\) |
Notes
- For the pdf’s above we understand \(f(x)=0\) for those \(x \in \R\) for which no value is specified. For example, in the case of a \(\text{Beta}(\alpha,\beta)\) distribution we have that \(f(x)=0\) for \(x \leq 0\) and for \(x \geq 1\).
- \(\Gamma\) denotes the Gamma function which can be defined as \[\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \d t \quad \text{for $x>0$.}\] Note that for integers \(x \in \{1,2,\ldots\}\) we have that \(\Gamma(x)=(x-1)!\).
- \(B\) denotes the Beta function which can be defined as \[B(x,y)=\int_0^1 t^{x-1} (1-t)^{y-1} \d t \quad \text{for $x,y>0$.}\] Note that for integers \(x,y \in \{1,2,\ldots\}\) we have that \[B(x,y) = \frac{(x-1)! (y-1)!}{(x+y-1)!}.\]