Binomial distribution

Binomial Experiment

A binomial experiment (or Bernoulli trial) is a statistical experiment that has the following properties:

The experiment consists of n repeated trials.
The trials are independent.
The outcome of each trial is either success (s) or failure (f).

Binomial Distribution

We define a binomial process to be a binomial experiment meeting the following conditions:

The number of successes is $x$.
The total number of trials is $n$.
The probability of success of 1 trial is $p$.
The probability of failure of 1 trial $q$, where $q=1-p$.
$b(x,n,p)$ is the binomial probability, meaning the probability of having exactly $x$ successes out of $n$ trials.

The binomial random variable is the number of successes, $x$, out of $n$ trials.

The binomial distribution is the probability distribution for the binomial random variable, given by the following probability mass function:

$$b(x,n,p) = \frac{n!}{x!(n-x)!}p^xq^{n-x}$$

Python code for the Binomial distribution

import math

def bi_dist(x, n, p):
    b = (math.factorial(n)/(math.factorial(x)*math.factorial(n-x)))*(p**x)*((1-p)**(n-x))
    return(b)

Using numpy

import numpy as np

n = 10 #number of coin toss
p = 0.5 #probability
samples = 1000 #number of samples
s = np.random.binomial(n, p, samples)

Using the stats module from scipy

from scipy.stats import binom

n = 10 #number of coin toss
p = 0.5 #probability
samples = 1000 #number of samples
s = binom.rvs(n, p, size=samples)

Cumulative probabilities

A cumulative probability refers to the probability that the value of a random variable falls within a specified range. Frequently, cumulative probabilities refer to the probability that a random variable is less than or equal to a specified value.

A fair coin is tossed 10 times.

Probability of getting 5 heads

The probability of getting heads is:

$$b(x=5, n=10, p=0.5)=0.246$$

Probability of getting at least 5 heads

The probability of getting at least heads is:

$$b(x\geq 5, n=10, p=0.5)= \sum_{r=5}^{10} b(x=r, n=10, p=0.5)$$

$$b(x\geq 5, n=10, p=0.5)= 0.623$$

Probability of getting at most 5 heads

The probability of getting at most heads is:

$$b(x\leq 5, n=10, p=0.5)= \sum_{r=0}^{5} b(x=r, n=10, p=0.5)$$

$$b(x\leq 5, n=10, p=0.5)= 0.623$$