STAT: Probability Distributions

Distribution		Constraints	Density Function	Expectation	Variance	MGF
			$f(x)$	$E[X] = \int x f(x) dx$	$\text{Var}(X) = E[(X-\mu)^2]$	$M_X(t) = E[e^{Xt}] = \int e^{xt} f(x)dx$
Discrete distributions
Poisson	$X \sim \text{Po}(\lambda)$	$k=0,1,2,\cdots$	$P(X=k) = \frac{\lambda^k}{k!}e^{-\lambda}$	$\lambda$	$\lambda$	$e^{\lambda(e^t-1)}$
Continuous distributions
Uniform	$X \sim U[a,b]$	$-\infty < x < \infty, a<b$	$f(x) = \frac{1}{b-a}\mathbf{1}_{[a,b]}(x)$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Standard Normal	$Z \sim \mathcal{N}(0,1)$	$-\infty < z < \infty$	$f_Z(z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}z^2}$	0	1	$e^{\frac{1}{2}t^2}$
Normal	$X \sim \mathcal{N}(\mu,\sigma^2)$	$-\infty < x < \infty$	$f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}$	$\mu$	$\sigma^2$	$e^{\mu t+\frac{1}{2}\sigma^2 t^2}$
Double Exponential		$-\infty < x < \infty, \lambda > 0$	$f_X(x) = \frac{\lambda}{2} e^{-\lambda\vert x \vert}$	0	$\frac{2}{\lambda^2}$
One sided distributions
Gamma	$X \sim G(\alpha,\beta)$	$x > 0, \alpha, \beta > 0$	$f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1}e^{-\beta x}$	$\frac{\alpha}{\beta}$	$\frac{\alpha}{\beta^2}$	$\left(\frac{\beta}{\beta-t}\right)^\alpha$
Exponential	$X \sim \exp(\beta) = G(1,\beta) = W(\frac{1}{\beta},1)$		$f(x) = \beta e^{-\beta x}$	$\frac{1}{\beta}$	$\frac{1}{\beta^2}$	$\left(\frac{\beta}{\beta-t}\right)$
Chi-Square	$X \sim \chi^2(\nu) = G(\frac{\nu}{2},\frac{1}{2})$		$f(x) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)} x^{\nu/2-1}e^{-x/2}$	$\nu$	$2\nu$	$\left(\frac{1}{1-2t}\right)^{\nu/2}$
Log Gamma (Heavy Tailed)	$Y = e^X \sim LG(\alpha,\beta), X \sim G(\alpha,\beta)$	$y > 1, \alpha, \beta > 0$	$f_Y(y) = \frac{\beta^\alpha}{\Gamma(\alpha)}(\log y)^{\alpha-1}y^{-(\beta+1)}$	$\left(\frac{\beta}{\beta-1}\right)^\alpha$	$\left(\frac{\beta}{\beta-2}\right)^\alpha - \left(\frac{\beta}{\beta-1}\right)^{2\alpha}$
Inverse Gamma (Heavy Tailed)	$Y = 1/X \sim IG(\alpha,\beta), X \sim G(\alpha,\beta)$	$y > 0, \alpha, \beta > 0$	$f_Y(y) = \frac{\beta^\alpha}{\Gamma(\alpha)}y^{-(\alpha+1)}e^{-\beta/y}$	$\left(\frac{\beta}{\alpha-1}\right)$	$\left(\frac{\beta^2}{(\alpha-2)(\alpha-1)^2}\right)$
Log Normal (Heavy Tailed)	$Y = e^X \sim LN(\mu,\sigma^2), X \sim \mathcal{N}(\mu,\sigma^2)$	$y > 0$	$f_Y(y) = \frac{1}{\sqrt{2\pi\sigma^2}}\frac{1}{y}e^{-\frac{1}{2\sigma^2}(\log y - \mu)^2}$
Weibull	$X \sim W(a,\gamma)$	$x > 0, a, \gamma > 0$	$f_X(x) = \frac{\gamma}{a^\gamma} x^{\gamma-1} e^{-(x/a)^\gamma}$
Pareto – 2 Parameters (Heavy Tailed)	$X \sim P(\alpha, \theta)$	$x > 0, \alpha, \theta > 0$	$f(x) = \frac{\alpha \theta^\alpha}{(\theta+x)^{1+\alpha}}$	$\frac{\theta}{\alpha-1}$	$\frac{2\theta^2}{(\alpha-1)(\alpha-2)}$
Pareto – 3 Parameters (Heavy Tailed)	$X \sim P(\alpha, \beta, \theta)$	$x > 0, \alpha, \beta, \theta > 0$	$f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\frac{\theta^\alpha x^{\beta-1}}{(\theta+x)^{\alpha+\beta}}$	$\frac{\beta\theta}{\alpha-1}$	$\frac{\beta(1+\beta)\theta^2}{(\alpha-1)(\alpha-2)}$