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 \(\mathbb{E}[\mathrm{Pois}(\lambda)]=\mathbb{V}\mathrm{ar}[\mathrm{Pois}(\lambda)]=\lambda.\)