# Distributions

### Gumbel Distribution

Used to model the extreme deviations from the median of distributions. Given the extreme(max/min) samples from an order statistic, models the distribution of these extreme values.

#### Model

$\alpha$ = the location parameter

$\beta$ = the scale parameter

#### Mean

$\mu=\alpha-\gamma\beta$ where $\gamma=lim_{n\to\infty}(\sum_{k=1}^n\frac{1}{k}-ln n)$, a.k.a Euler’s constant

#### Application

[1] Given the data for past 10 years’ highest earthquake on richter scale, model

[2] TODO: The maximum value in a sample of RVs coming from exponential distribution can be modelled using Gumbel distribuion