# Statistical Distribution

• Discrete distribution
• Contious distribution
• Sample(small) distribution

# Discrete Distribution

• Binomial distribution $B(n,p)$
• Hypergeometric distribution
• Geometric distribution
• Poisson distribution $P(\lambda)$

## 1.1 Binomial distribution $B(n,p)$

$$P(X=k) = C_n^k p^k (1-p)^{n-k}, k=0,1,...,n$$

then $X \sim B(n,p)$.



In [1]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

n,p=50,0.1

plt.hist(np.random.binomial(n,p,size=5000))
plt.show()






## 1.2 Hypergeometric distribution

$$P(X=k) = \frac {C_M^k C_{N-M}^{n-k}} {C_N^n}$$

then $X \sim$ Hypergeometric distribution with parameters $\{N,M,n\}$



In [2]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

ngood, nbad, nsamp = 90, 10, 50

plt.show()






## 1.3 Geometric distribution

$$P(X=k) = p(1-p)^{k-1}$$


In [3]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.geometric(p=0.35, size=10000))
plt.show()






## 1.4 Poisson distribution $P(\lambda)$

$$P(X=k) = e^{-\lambda} \frac{\lambda^k}{k!}, k = 0,1,2,...$$

then $X \sim P(\lambda)$, $\lambda>0$



In [4]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.poisson(5, 10000))
plt.show()






# Contious distribution

• Homogeneous distribution $R(a,b)$
• Exponential distribution $E(\lambda)$
• Normal distribution $N(\mu,\sigma^2)$

## 2.1 Homogeneous distribution $R(a,b)$

$$f(x)=\left\{\begin{array}{ll} c, & a<x<b \\ 0, & otherwise \end{array}\right.$$

then $X \sim R(a,b)$



In [5]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.random_sample(1000))
plt.show()






## 2.2 Exponential distribution $E(\lambda)$

$$f(x)=\left\{\begin{array}{ll} \lambda e^{-\lambda x}, & x>0 \\ 0, & otherwise \end{array}\right.$$

then $X \sim E(\lambda)$



In [6]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.exponential(scale=1.0, size=1000))
plt.show()






## 2.3 Normal distribution $N(\mu,\sigma^2)$

$$f(x) = \frac {1} {\sqrt{2\pi\sigma}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

then $X \sim N(\mu,\sigma^2)$.



In [7]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.normal(size=4000))
plt.show()






# Sample(small) distribution

• $\chi^2$ distribution $\chi^2(n)$
• t distribution $t(n)$
• F distribution $F(n)$

## 3.1 $\chi^2$ distribution $\chi^2(n)$

Given $X_i \sim N(0,1)$, $$Y = \sum_{i=1}^{N} X_i^2 \sim \chi^2(n)$$



In [8]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.chisquare(3,1000))
plt.show()






## 3.2 t distribution $t(n)$

Given $X \sim N(0,1)$, $Y \sim \chi^2(n)$, $$T \hat{=} \frac{X}{\sqrt{\frac{Y}{n}}} \sim t(n)$$



In [9]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.standard_t(2,50))
plt.show()






## 3.3 F distribution $F(n)$

Given $X \sim \chi^2(m)$, $Y \sim \chi^2(n)$, $$F \hat{=} \frac{\frac{X}{m}}{\frac{Y}{n}} \sim F(m,n)$$



In [10]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.hist(np.random.f(4,10,5000))
plt.show()







In [ ]: