Posted on Aug 4, 2024

USE AND ENJOY THE BINOMIAL DISTRIBUTION MODEL

#inferential #statistics #binomial #discrete

In statistics and probability theory, the term “binomial” often refers to the binomial distribution, which is a discrete probability distribution.

𝑃(𝑋=𝑥)=(𝑛𝑥)𝑝𝑥(1−𝑝)𝑛−𝑥

A binomial is a mathematical expression consisting of two terms connected by a plus or minus sign. In algebra, a simple example of a binomial is a + b, where a and b are terms that can represent numbers, variables, or more complex expressions.

The binomial distribution models the number of successes in a fixed number of independent trials of a binary experiment (an experiment with two possible outcomes: success or failure). Each trial has the same probability of success, denoted by p.

Key properties of a binomial distribution include:

Number of Trials (n): The fixed number of independent trials.
Probability of Success (p): The probability of success on a single trial.
Probability of Failure (q): The probability of failure on a single trial, where q = 1 - p.
Random Variable (X): The number of successes in n trials.

 import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns import scipy.stats as stats from scipy.stats import binom %matplotlib inline # Standard notation # P = binomial probability # x = number of times for a specific outcome within n trials # n = number of trials # p = probability of success on a single trial # q = probability of failure on a single trial # k = an array of n, the number of trials # p_of_k = probabilty of successes for each trial # am = at most = Define at most successes # Probability mass function = .pmf() # Probability density function = .pdf() # The probability mass function of the binomial distribution is # f(x)=P[X=x]=(nx)px(1−p)n−x n = (10) p = (0.8) k = np.arange(0,11) am = (6) p_of_k = binom.pmf(n,n,p) p_of_k x = binom.pmf(k,n,p) x barl = plt.bar(k, x, color='hotpink') plt.title(('When p = ' + str(p) + ' and at most ' + str(am) + ' successes'), fontsize=13, color='r') plt.legend((p, ''), fontsize=10) plt.xlabel('Number of Successes', fontsize=10, color='r') plt.ylabel('Probability of Successes', fontsize=10, color='royalblue') for i in range(0, am): barl[i].set_color('r')