# Distribution

For each distribution, we try to figure out:

- What is its support?
- What is the p.m.f.?
- What are the parameters?
- What is $E(X)$? What is $Var(X)$ and s.d.?
- Why is it important?

### Discrete vs. Continuous Distributions

Discrete | Continuous | |
---|---|---|

Expected Value | $E(x)=βxf(x)$ | $E(x)=β«_{ββ}x(f(x))dx$ |

Variance | $Var(x)=E(x_{2})β(E(x))_{2}$ | $Var(x)=β«_{ββ}x_{2}f(x)dxβ[β«_{ββ}x(f(x)]_{2}$ |

pmf to cdf | $F(x)=β_{y:yβ€x}f(y)$ | $F(x)=β«_{ββ}f(y)dy$ |

cdf to pdf | $f(x)=F(x)βF(xβ1)$ | $dxdFβ=f(x)$ |

### Distributions

- Bernoulli Distribution
- Binomial Distribution
- Geometric Distribution
- Negative Binomial Distribution
- Normal Distribution
- Hypergeometric Distribution
- Poisson Distribution
- Multinomial Distribution
- Cauchy-Lorentz Distribution

### Concepts

Measures of dispersion and symmetry

- Range = max - min
- IQR (inter-quartile range) = $Q_{3}βQ_{1}$
- Variance (standard dev), $s_{2}=nβ11ββ(x_{i}βx)_{2}$
- Skewness - measures bias towards left or right side
- Kurtosis - measure of normality

### 1D Distribution

I was having trouble understanding what a 1D Distribution was, since I needed to use that to calculate the Wasserstein Metric. Um, I think itβs just with one variable.

### 2D Distribution

This is where we talk about multivariate distributions. https://en.wikipedia.org/wiki/Multivariate_normal_distribution

### Distance between Distributions

How do you quantify how far away two distributions are? There seems to be quite a few methods, that I found from stackoverflow

I ran into this problem while working on my Poker AI, since Euclidean distance is not the best measure.

Heuristic

- Minkowski-form
- Weighted-Mean-Variance (WMV)

Nonparametric test statistics

- 2 (Chi Square)
- Kolmogorov-Smirnov (KS)
- Cramer/von Mises (CvM)

Information-theory divergences

- KL Divergence
- JensenβShannon divergence (metric)
- Jeffrey-divergence (numerically stable and symmetric)

Ground distance measures

- Histogram intersection
- Quadratic form (QF)
- Earth Moverβs Distance