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')
The binomial distribution is widely used in various fields, including biology, finance, and engineering, to model binary outcomes.
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