Summary of parametric distributions

Here a summary of the main equations for each of the presented distirbution functions is presented.

Distribution	PDF	CDF	Mean and variance
Normal/Gaussian	\(f(x) = \cfrac{1}{\sigma \sqrt{2\pi}}e^{\left(\normalsize-\cfrac{(x-\mu)^2}{2\sigma^2}\right)}\)	\(F(x) = \cfrac{1}{2}\left(1+\text{erf}\left(\cfrac{x-\mu}{\sigma\sqrt{2}}\right)\right)\)	\(\begin{array}{ll} E[X] = \mu \\ Var[X] = \sigma^2 \end{array}\)
Uniform	\(\displaystyle f(x) = \begin{cases}\cfrac{1}{b-a} & \text{for }x \in [a,b] \\ 0 & \text{otherwise} \end{cases}\)	\(F(x)=\begin{cases}0 & \text{for } x<a \\ \cfrac{x-a}{b-a} & \text{for } x\in[a,b] 1 & \text{for } x>b\end{cases}\)	\(\begin{array}{ll} E[X]=\frac{1}{2}(a+b) \\ Var[X]=\frac{1}{12}(b-a)^2 \end{array}\)
Exponential	\(f(x) = \lambda e^{\normalsize-\lambda x}\)	\(F(x) = 1 - e^{\normalsize-\lambda x}\)	\(\begin{array}{ll} E[X] = \cfrac{1}{\lambda} \\ Var[X] = \cfrac{1}{\lambda^2} \end{array}\)
Gumbel	\(f(x) = \cfrac{1}{\beta} e^{\normalsize-\left(z + e^{\normalsize-z}\right)}\text{, where }z=\cfrac{x-\alpha}{\beta}\)	\(F(x)=e^{\normalsize-e^{\normalsize-z}}\)	\(\begin{array}{ll} E[X] = \alpha + \beta\gamma,\; \gamma = 0.5772 \\ Var[X] = \cfrac{\pi^2}{6}\beta^2 \end{array}\)
Lognormal	\(f(x) = \cfrac{1}{x \sigma \sqrt{2 \pi}}e^{\left( \normalsize-\cfrac{(ln(x)-\mu)^2}{2\sigma^2}\right)}\)	\( F(x) = \Phi\left( \cfrac{ln(x)-\mu}{\sigma} \right) = \frac{1}{2}\left[ 1+\text{erf}\left( \cfrac{ln(x)-\mu}{\sigma \sqrt{2}}\right)\right] \)	\(\begin{array}{ll} E[X]=e^{\normalsize\mu + \frac{\sigma^2}{2}} \\ Var[X] = \left( e^{\normalsize\sigma^2}-1 \right)e^{2\mu + \sigma^2} \end{array}\)

MUDE exam information

You do not need to know the equations of the distribution functions by heart. You just need to know how the distribution looks (PDF/CDF), how it responds to changes in the parameters and some basic properties (symmetry or bounds).

Attribution

This chapter was written by Patricia Mares Nasarre and Robert Lanzafame. Find out more here.