MCS: continuous random variable - student's t distribution

Student’s t

Student’s t Distribution is also a very important distribution in statistical analysis , It is often used to test the significance of the mean value of variables .Student’s t Distribution is also called t Distribution , Similar to the standard normal distribution , But the tail can extend to the left and right , Depends on the parameters $k$ freedom .

t Expectation and variance of distribution ：

$$E(t)=0$$
$$V(t)=\frac{k}{k-2},k>2$$

t Distribution properties ：

1. When $k>30$ when ,t The distribution approximates the standard normal distribution .
2. t、chi-Square And the standard normal distribution ：
   $$t=\frac{z}{\sqrt{{\mathcal{X}}^2/k}}$$

Generative obedience t Random variable of distribution

1. Generate a standard normal variable ：$z\sim N(0,1)$
2. Generate a degree of freedom k Of Chi-Square Variable ：${\mathcal{X}}_k^2$
3. $t=z/\sqrt{{\mathcal{X}}_k^2/k}$
4. Return t.

example ： Generate a degree of freedom 6 Of t Variable ：

1. $z=0.71$
2. ${\mathcal{X}}_6^2=6.29$
3. $t=0.71/\sqrt{6.29/6}=0.693$
4. $t=0.693$

import numpy as np
import matplotlib.pyplot as plt

def generate_t_var(k=1):
    z = np.random.normal(0, 1)
    if k < 30:
        c = (np.random.normal(0, 1, size=(k))**2).sum()
        t = z / np.sqrt(c / k)
    else:
        z_ = np.random.normal(0, 1)
        c = int(k + z_ * np.sqrt(2*k) + 0.5)
        t = z / np.sqrt(c / k)
    return t

$k=1、10、30、50$

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