1.2 Some facts about distributions

We will make use of several parametric distributions. Some notation and facts are introduced as follows:

• $\mathcal{N}(\mu,\sigma^2)$ stands for the normal distribution with mean $\mu$ and variance $\sigma^2.$ Its pdf is $\phi_\sigma(x-\mu):=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}},$ $x\in\mathbb{R},$ and satisfies that $\phi_\sigma(x-\mu)=\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)$ (if $\sigma=1$ the dependence is omitted). Its cdf is denoted as $\Phi_\sigma(x-\mu).$ $z_\alpha$ denotes the upper $\alpha$-quantile of a $\mathcal{N}(0,1),$ i.e., $z_\alpha=\Phi^{-1}(1-\alpha).$ Some uncentered moments of $X\sim\mathcal{N}(\mu,\sigma^2)$ are

  $$\begin{align*} \mathbb{E}[X]&=\mu,\\ \mathbb{E}[X^2]&=\mu^2+\sigma^2,\\ \mathbb{E}[X^3]&=\mu^3+3\mu\sigma^2,\\ \mathbb{E}[X^4]&=\mu^4+6\mu^2\sigma^2+3\sigma^4. \end{align*}$$

  The multivariate normal is represented by $\mathcal{N}_p(\boldsymbol{\mu},\boldsymbol{\Sigma}),$ where $\boldsymbol{\mu}$ is a $p$-vector and $\boldsymbol{\Sigma}$ is a $p\times p$ symmetric and positive matrix. The pdf of a $\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ is $\phi_{\boldsymbol{\Sigma}}(\mathbf{x}-\boldsymbol{\mu}):=\frac{1}{(2\pi)^{p/2}|\boldsymbol{\Sigma}|^{1/2}}e^{-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})'\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})},$ and satisfies that $\phi_{\boldsymbol{\Sigma}}(\mathbf{x}-\boldsymbol{\mu})=|\boldsymbol{\Sigma}|^{-1/2}\phi\left(\boldsymbol{\Sigma}^{-1/2}(\mathbf{x}-\boldsymbol{\mu})\right)$ (if $\boldsymbol{\Sigma}=\mathbf{I}$ the dependence is omitted).

• The lognormal distribution is denoted by $\mathcal{LN}(\mu,\sigma^2)$ and is such that $\mathcal{LN}(\mu,\sigma^2)\stackrel{d}{=}\exp(\mathcal{N}(\mu,\sigma^2)).$ Its pdf is $f(x;\mu,\sigma):=\frac{1}{\sqrt{2\pi}\sigma x}\allowbreak e^{-\frac{(\log x-\log\mu)^2}{2\sigma^2}},$ $x>0.$ Note that $\mathbb{E}[\mathcal{LN}(\mu,\sigma^2)]=e^{\mu+\frac{\sigma^2}{2}}$

• The exponential distribution is denoted as $\mathrm{Exp}(\lambda)$ and has pdf $f(x;\lambda)=\lambda e^{-\lambda x},$ $\lambda,x>0.$

• The gamma distribution is denoted as $\Gamma(a,p)$ and has pdf $f(x;a,p)=\frac{a^p}{\Gamma(p)} x^{p-1}e^{-a x},$ $a,p,x>0,$ where $\Gamma(p)=\int_0^\infty x^{p-1}e^{-ax}\,\mathrm{d}x.$ It is known that $\mathbb{E}[\Gamma(a,p)]=\frac{p}{a}$ and $\mathbb{V}\mathrm{ar}[\Gamma(a,p)]=\frac{p}{a^2}.$

• The inverse gamma distribution, $\mathrm{IG}(a,p)\stackrel{d}{=}\Gamma(a,p)^{-1},$ has pdf $f(x;a,p)=\frac{a^p}{\Gamma(p)} x^{-p-1}e^{-\frac{a}{x}},$ $a,p,x>0.$ It is known that $\mathbb{E}[\mathrm{IG}(a,p)]=\frac{a}{p-1}$ and $\mathbb{V}\mathrm{ar}[\mathrm{IG}(a,p)]=\frac{a^2}{(p-1)^2(p-2)}.$

• The binomial distribution is denoted as $\mathrm{B}(n,p).$ Recall that $\mathbb{E}[\mathrm{B}(n,p)]=np$ and $\mathbb{V}\mathrm{ar}[\mathrm{B}(n,p)]=np(1-p).$ A $\mathrm{B}(1,p)$ is a Bernoulli distribution, denoted as $\mathrm{Ber}(p).$

• The beta distribution is denoted as $\beta(a,b)$ and its pdf is $f(x;a,b)=\frac{1}{\beta(a,b)}x^{a-1}(1-x)^{1-b},$ $0<x<1,$ where $\beta(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.$ When $a=b=1,$ the uniform distribution $\mathcal{U}(0,1)$ arises.

• The Poisson distribution is denoted as $\mathrm{Pois}(\lambda)$ and has pdf $\mathbb{P}[X=x]=\frac{x^\lambda e^{-\lambda}}{x!},$ $x=0,1,2,\ldots.$ Recall that $\mathbb{E}[\mathrm{Pois}(\lambda)]=\mathbb{V}\mathrm{ar}[\mathrm{Pois}(\lambda)]=\lambda.$