Examples of probability distribution operations implemented in Python

import numpy as np
from scipy import stats
import  as plt
'''
# Binomial distribution
# Prerequisites: independently repeated trials, with playbacks, only two outcomes
# The binomial distribution states that the probability of an event A occurring on a randomized trial, if p, is the probability of an event A occurring k times in n repetitions of the trial:
# f(n,k,p) = choose(n, k) * p**k * (1-p)**(n-k)
'''
# ① Define the basic information of the binomial distribution
p = 0.4 # Event A probability 0.4
n = 5  # Repeat the experiment 5 times
k = (n+1) # 6 possible outcomes
#k = ((0.01,n,p), (0.99,n,p), n+1) #anotherway
# ② Calculate the probability mass function of the binomial distribution.
# The reason it's called mass is that discrete points with a default volume (i.e. width) of 1
# P(X=x) --> is the probability that
probs = (k, n, p)
#array([ 0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024])
#(k, probs)
# ③ Calculate the cumulative density function of the binomial distribution.
# P(X<=x) --> also probabilities
cumsum_probs = (k, n, p)
#array([ 0.07776, 0.33696, 0.68256, 0.91296, 0.98976, 1.   ])
# ④ According to the cumulative probability to get the corresponding k, here lazy, directly use the above cumsum_probs
k2 = (cumsum_probs, n, p)
#array([0, 1, 2, 3, 4, 5])
# ⑤ Fake random variates with binomial distribution.
X = (n,p,size=20)
#array([2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 3, 0, 1, 1, 1, 2, 3, 4, 0, 3])
#⑧ Make a histogram of the frequency of random variables that satisfy the binomial distribution above (similar to group by).
(X)
#9 Make a histogram of the frequency distribution of the random variable satisfying the binomial distribution above.
(X, normed=True)
()

'''
Standard normal distribution
Density function: f(x) = exp(-x**2/2)/sqrt(2*pi)
'''
x = ((0.01), (0.99), 100)
# Probability density function
# It's called a density because of the continuous points, which have a default volume of 0
# f(x) --> not probability
probs = (x)
# (x, probs, 'r-', lw=5, alpha=0.6, label='norm pdf')
# Cumulative density function Cumulative density function
# Definite integral ∫_-oo^a f(x)dx --> is the probability of
cumsum_probs = (x)
# Falsify a random variable X that conforms to a normal distribution
# The loc and scale parameters allow you to specify the offset and scaling parameters of a random variable. For a normally distributed random variable, these two parameters are equivalent to specifying its expected value and standard deviation:
X = (loc=1.0, scale=2.0, size=1000)
#9 Make a histogram of the frequency distribution of the normally distributed random variable above.
(X, normed=True, histtype='stepfilled', alpha=0.2)
(loc='best', frameon=False)
()
# Parameter estimation for the given data. Being lazy here, I'll just use the above X
mean, std = (X)
#array(1.01810091), array(2.00046946)