A random variable $X$ has a continuous distribution if its sample space is uncountable.
Uniform Distribtution
\[f(x\vert a,b) = \begin{cases} \frac{1}{b-a} & \text{if $ x \in [a,b]$} \\ 0 & \text{otherwise} \end{cases}\]| \[\E(X)\] | \[\frac{b+a}{2}\] |
| \[Var(X)\] | \[\frac{(b-a)^2}{12}\] |
| \[M_X(t)\] | \[\frac{e^{tb}-e^{ta}}{t(b-a)}\] |
Exponential Distribution
$X \sim Exp(\lambda).$ With rate parameter $\lambda.$
\[f(x\vert \lambda) = \begin{cases} \lambda e^{-\lambda x}, & x\geq 0 \\ 0, & \text{else} \end{cases}\]| \[\E(X)\] | \[\frac{1}{\lambda}\] |
| \[Var(X)\] | \[\frac{1}{\lambda^2}\] |
| \[M_X(t)\] | \[(1-t\lambda^{-1})^{-1}\] |
Sometimes the exponential distribution is described with a scale parameter, $\beta$, instead where $\beta=1/\lambda.$
\[f(x\vert \beta) = \begin{cases} \frac{1}{\beta} e^{-x/\beta}, & x\geq 0 \\ 0, & \text{else} \end{cases}\]Gamma Distribution
Gamma function: \[\Gamma(\alpha) = \int_0^{\infty} t^{\alpha-1}e^{-t}\,dt.\] Properties: \[\Gamma(\alpha+1) = \alpha\Gamma(\alpha), \quad \alpha>0\] \[\Gamma(n) = (n-1)! \quad \text{if $n$ is an integer}\] $X \sim Gamma(\alpha, \beta).$
\[f(x \vert \alpha, \beta) = \frac{1}{\Gamma(\alpha)\beta^{\alpha}} x^{\alpha-1}e^{-x/\beta}, \quad 0 < x < \infty, \quad \alpha > 0, \quad \beta > 0.\]
| \[EX\] | \[\alpha\beta\] |
| \[Var\,X\] | \[\alpha\beta^2\] |
| \[M_X(t)\] | \[\left(\frac{1}{1-\beta t}\right)^{\alpha}, \quad t< \frac{1}{\beta}\] |
Note: $X\sim Exp(\beta)$ is a special case of the gamma distribution when $\alpha=1.$
Weibull Distribution
If $Y \sim Exp(\beta),$ then $X=Y^{1/\gamma} \sim Weibull(\gamma,\beta).$
\[f_X(x \vert \gamma,\beta) = \frac{\gamma}{\beta}x^{\gamma-1}e^{-x^{\gamma}/\beta}, \quad 0 < x < \infty, \quad \gamma>0, \quad \beta>0.\] However, a reparameterization of $\beta$ gives us the more common pdf: \[f_X(x \vert \gamma,\beta) = \frac{\gamma}{\beta}\left(\frac{x}{\beta}\right)^{\gamma-1}e^{-(x/\beta)^{\gamma}}, \quad 0 < x < \infty, \quad \gamma>0, \quad \beta>0.\]
| \[EX\] | \[\beta\Gamma\left(1+\frac{1}{\gamma}\right)\] |
| \[Var\,X\] | \[\beta^2\left[\Gamma\left(1+\frac{2}{\gamma}\right)-\left(\Gamma^2\left(1+\frac{1}{\gamma}\right)\right)\right]\] |
| \[M_X(t)\] | \[\sum_{n=0}^{\infty} \frac{(t\beta)^n}{n!}\Gamma\left(1+\frac{n}{\gamma}\right),\quad \gamma\geq 1\] |
Chi-Squared Distribution
Another special case of the gamma distribution is when $\alpha=p/2$ where $p$ is an integer and $\beta=2$ which has pdf \[f(x\vert p) = \frac{1}{\Gamma\left(\frac{p}{2}\right)2^{p/2}}x^{p/2-1}e^{-x/2}, \quad 0 < x < \infty\] which is called the chi-squared distribution with $p$ degrees of freedom. We write $X \sim \chi_p^2.$
| \[EX\] | \[p\] |
| \[Var\,X\] | \[2p\] |
| \[M_X(t)\] | \[(1-2t)^{-p/2}, \quad t<\frac{1}{2}\] |
Normal Distribution
$X \sim N(\mu, \sigma^2)$
\[f(x \vert \mu, \sigma^2) = \frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2/(2\sigma^2)}, \quad -\infty < x < \infty\]
| \[EX\] | \[\mu\] |
| \[Var\,X\] | \[\sigma^2\] |
| \[M_X(t)\] | \[\exp\left(\mu t + \sigma^2 t^2/2\right)\] |
If $X\sim N(\mu,\sigma^2),$ then the random variable $Z=\displaystyle\frac{X-\mu}{\sigma}$ has an $N(0,1)$ distribution and is called the standard normal.
Beta Distribution
$X\sim Beta(\alpha,\beta).$
\[f(x \vert \alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad 0< x < 1, \quad \alpha > 0\, \quad \beta > 0\] where $B(\alpha, \beta)$ denotes the beta function: \[B(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}.\]
| \[EX\] | \[\frac{\alpha}{\alpha+\beta}\] |
| \[Var\,X\] | \[\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\] |
| \[M_X(t)\] | \[1 + \sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r}\right)\frac{t^k}{k!}\] |
Cauchy Distribution
$X\sim Cauchy(\mu,\sigma)$
\[f(x\vert\mu,\sigma) = \frac{1}{\sigma\pi\left(1+\left(\frac{x-\mu}{\sigma}\right)^2\right)}, \quad -\infty < x < \infty.\]
| \[EX\] | Undefined |
| \[Var\,X\] | Undefined |
| \[M_X(t)\] | DNE |
Lognormal Distribution
If $X$ is a random variable whose logarithm is normally distributed (ie, $\ln X \sim N(\mu,\sigma^2)$), then $X$ has a lognormal distribution.
$X\sim Lognormal(\mu,\sigma^2).$
\[f(x \vert \mu,\sigma^2) = \frac{1}{\sqrt{2\pi}\sigma x}e^{-(\ln x-\mu)^2/(2\sigma^2)},\quad 0 < x < \infty, \quad -\infty < \mu < \infty, \quad \sigma>0 \]
| \[EX\] | \[e^{\mu + (\sigma^2/2)}\] |
| \[Var\,X\] | \[e^{2(\mu+\sigma^2)}-e^{2\mu+\sigma^2}\] |
| \[M_X(t)\] | \[??\] |
Double Exponential Distribution
$X \sim DoubleExp(\mu\sigma).$
\[f(x\vert\mu,\sigma) = \frac{1}{2\sigma}e^{-|x-\mu|/\sigma},\quad -\infty < x < \infty,\quad -\infty < \mu < \infty, \quad \sigma>0\]
| \[EX\] | \[\mu\] |
| \[Var\,X\] | \[2\sigma^2\] |
| \[M_X(t)\] | \[\frac{e^{\mu t}}{1-\sigma^2t^2},\quad |t|<\frac{1}{\sigma}\] |