Mean

2026-01-06

Statistics

2025-08-22 (updated: 2026-01-06)
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This page pairs well with Probability.

Table of Distributions

Distributionmass/density function\[S_X\]\[\mathbb{E}(X)\]\[\mathrm{Var}(X)\]\[\phi_X(s)\]
Bernoulli \[\mathrm{Bern}(\pi)\]\[P(X=1)=\pi\\ P(X=0)=1-\pi\]\[\{0,1\}\]\[\pi\]\[\pi(1-\pi)\]\[(1-\pi)+\pi e^{s}\]
Binomial \[\mathrm{Bin}(n,\pi)\]\[p_X(x)=\binom{n}{x}\pi^{x}(1-\pi)^{n-x}\]\[\{0,1,\dots,n\}\]\[n\pi\]\[n\pi(1-\pi)\]\[(1-\pi+\pi e^{s})^{n}\]
Geometric \[\mathrm{Geo}(\pi)\]\[p_X(x)=\pi(1-\pi)^{x-1}\]\[\{1,2,\dots\}\]\[\pi^{-1}\]\[(1-\pi)\pi^{-2}\]\[\frac{\pi}{e^{-s}-1+\pi}\]
Poisson \[\mathcal{P}(\lambda)\]\[p_X(x)=e^{-\lambda}\lambda^{x}/x!\]\[\{0,1,\dots\}\]\[\lambda\]\[\lambda\]\[\exp\{\lambda(e^{s}-1)\}\]
Uniform \[U[\alpha,\beta]\]\[f_X(x)=(\beta-\alpha)^{-1}\]\[[\alpha,\beta]\]\[\frac{1}{2}(\alpha+\beta)\]\[\frac{1}{12}(\beta-\alpha)^2\]\[\frac{e^{\beta s}-e^{\alpha s}}{s(\beta-\alpha)}\]
Exponential \[\mathrm{Exp}(\lambda)\]\[f_X(x)=\lambda e^{-\lambda x}\]\[[0,\infty)\]\[\lambda^{-1}\]\[\lambda^{-2}\]\[\frac{\lambda}{\lambda-s}\]
Gaussian \[\mathcal{N}(\mu,\sigma^{2})\]\[f_X(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^{2}}\right\}\]\[\mathbb{R}\]\[\mu\]\[\sigma^{2}\]\[e^{\mu s+\frac{1}{2}\sigma^{2}s^{2}}\]
Gamma \[\Gamma(\alpha,\lambda)\]\[f_X(x)=\frac{1}{\Gamma(\alpha)}\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}\]\[[0,\infty)\]\[\alpha\lambda^{-1}\]\[\alpha\lambda^{-2}\]\[\left(\frac{\lambda}{\lambda-s}\right)^{\alpha}\]

Statistical Inference

Definition (Random Sample & Model)

Let \(X=(X_1,\ldots,X_n)\) be i.i.d. from a parametric family \(\{F_\theta:\theta\in\Theta\subset\mathbb{R}^p\}\). The parameter \(\theta\) is unknown; inference uses the randomness of \(X\) to learn about \(\theta\).

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