# Related Probability Distributions

# Binomial distribution [discrete]

The binomial distribution is a discrete distribution that describes the probability of getting exactly x successes in n trials of a probability experiment:

where n = the total number of trials, x = the number of successes (1, 2, 3, …, n), p = the probability of success (e.g., getting head in coin flip; user click a link for web activity analysis)

## Example: Probability mass function of getting heads with 10 coin flips from 10000 repeated trials of the experiment

import scipy.stats# Simulate random variable for the number of heads obtainedr = scipy.stats.binom.rvs(n=10, p=0.5, size=10000, random_state=42)

# from 10 coin flips carried over 10000 repeated experiment trials

** Concept refresher: What is PMF? **PMF stands for probability mass function. It is the probability distribution function for discrete distributions and describes the probability associated with each discrete outcome.

## The mean and standard deviation of a binomial distribution

binomial_mean = n * p

# binomial_mean = 10 * 0.5 = 5binomial_std = np.sqrt(n * p * (1 - p))

# binomial_std = np.sqrt(10 * 0.5 * (1 -0.5)) = 1.58

# Normal distribution [continuous]

**Binomial approximation: large n**

Despite the continuous vs. discrete distinction between the binomial and normal distributions, **once the number of trials — that is, the parameter n — within each round of binomial experiment becomes large enough**, binomial distribution becomes a good approximation of the normal distribution.

`r = scipy.stats.binom.rvs(n=10000, p=0.5, size=10000, random_state=42)`

# Poisson distribution [discrete]

**Binomial approximation: large n, very low p**

Poisson distribution describes the probability of obtaining x number of occurrences over a certain period of time (e.g., sales of an item per day). It can be approximated by a binomial distribution with large size n and a small probability p:

`# Use low p binomial to approximate poisson distribution`

r = scipy.stats.binom.rvs(n=10000, p=0.5e-3, size=10000, random_state=42)

Whereas Poisson distribution has the properties of having the same mean and standard deviation value lambda, the ** negative binomial distribution** adjusts variance indepdently from the mean and thus allows more flexibility. In fact, the Poisson distribution is a special case of the negative biomial distribution.