Table of Distributions
Table of Distributions#
Distribution |
PMF/PDF and Support |
Expected Value |
Variance |
MGF |
|---|---|---|---|---|
{Bernoulli \ \Bern(\(p\))} |
{\(P(X=1) = p\) \\( P(X=0) = q=1-p\)} |
\(p\) |
\(pq\) |
\(q + pe^t\) |
{Binomial \ \Bin(\(n, p\))} |
{\(P(X=k) = {n \choose k}p^k q^{n-k}\) \ \(k \in \{0, 1, 2, \dots n\}\)} |
\(np\) |
\(npq\) |
\((q + pe^t)^n\) |
{Geometric \ \Geom(\(p\))} |
{\(P(X=k) = q^kp\) \ \(k \in \{\)0, 1, 2, \dots \(\}\)} |
\(q/p\) |
\(q/p^2\) |
\(\frac{p}{1-qe^t}, \, qe^t < 1\) |
{Negative Binomial \ \NBin(\(r, p\))} |
{\(P(X=n) = {r + n - 1 \choose r -1}p^rq^n\) \ \(n \in \{\)0, 1, 2, \dots \(\}\)} |
\(rq/p\) |
\(rq/p^2\) |
\((\frac{p}{1-qe^t})^r, \, qe^t < 1\) |
{Hypergeometric \ \Hypergeometric(\(w, b, n\))} |
{\(P(X=k) = \sfrac{{w \choose k}{b \choose n-k}}{{w + b \choose n}}\) \ \(k \in \{0, 1, 2, \dots, n\}\)} |
\(\mu = \frac{nw}{b+w}\) |
\(\left(\frac{w+b-n}{w+b-1} \right) n\frac{\mu}{n}(1 - \frac{\mu}{n})\) |
messy |
{Poisson \ \Pois(\(\lambda\))} |
{\(P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}\) \ \(k \in \{\)0, 1, 2, \dots \(\}\)} |
\(\lambda\) |
\(\lambda\) |
\(e^{\lambda(e^t-1)}\) |
{Uniform \ \Unif(\(a, b\))} |
{\( f(x) = \frac{1}{b-a}\) \\( x \in (a, b) \)} |
\(\frac{a+b}{2}\) |
\(\frac{(b-a)^2}{12}\) |
\(\frac{e^{tb}-e^{ta}}{t(b-a)}\) |
{Normal \ \(\N(\mu, \sigma^2)\)} |
{\(f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\sfrac{(x - \mu)^2}{(2 \sigma^2)}}\) \ \(x \in (-\infty, \infty)\)} |
\(\mu\) |
\(\sigma^2\) |
\(e^{t\mu + \frac{\sigma^2t^2}{2}}\) |
{Exponential \ \(\Expo(\lambda)\)} |
{\(f(x) = \lambda e^{-\lambda x}\)\\( x \in (0, \infty)\)} |
\(\frac{1}{\lambda}\) |
\(\frac{1}{\lambda^2}\) |
\(\frac{\lambda}{\lambda - t}, \, t < \lambda\) |
{Gamma \ \(\Gam(a, \lambda)\)} |
{\(f(x) = \frac{1}{\Gamma(a)}(\lambda x)^ae^{-\lambda x}\frac{1}{x}\)\\( x \in (0, \infty)\)} |
\(\frac{a}{\lambda}\) |
\(\frac{a}{\lambda^2}\) |
\(\left(\frac{\lambda}{\lambda - t}\right)^a, \, t < \lambda\) |
{Beta \ \Beta(\(a, b\))} |
{\(f(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}\)\\(x \in (0, 1) \)} |
\(\mu = \frac{a}{a + b}\) |
\(\frac{\mu(1-\mu)}{(a + b + 1)}\) |
messy |
{Log-Normal \ \(\mathcal{LN}(\mu,\sigma^2)\)} |
{\(\frac{1}{x\sigma \sqrt{2\pi}}e^{-(\log x - \mu)^2/(2\sigma^2)}\)\\(x \in (0, \infty)\)} |
\(\theta = e^{ \mu + \sigma^2/2}\) |
\(\theta^2 (e^{\sigma^2} - 1)\) |
doesn’t exist |
{Chi-Square \ \(\chi_n^2\)} |
{\(\frac{1}{2^{n/2}\Gamma(n/2)}x^{n/2 - 1}e^{-x/2}\)\\(x \in (0, \infty) \)} |
\(n\) |
\(2n\) |
\((1 - 2t)^{-n/2}, \, t < 1/2\) |
{Student-\(t\) \ \(t_n\)} |
{\(\frac{\Gamma((n+1)/2)}{\sqrt{n\pi} \Gamma(n/2)} (1+x^2/n)^{-(n+1)/2}\)\\(x \in (-\infty, \infty)\)} |
\(0\) if \(n>1\) |
\(\frac{n}{n-2}\) if \(n>2\) |
doesn’t exist |