Difference Between Binomial and Poisson Distribution (original) (raw)

Last Updated : 23 Jul, 2025

Binomial and Poisson distributions are two important types of discrete probability distributions used in statistics and data analysis. binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. On the other hand, the Poisson distribution models the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence.

In this article, we will discuss "Difference Between Binomial and Poisson Distribution" in detail, including properties and examples for each.

What is Binomial Distribution?

**Binomial Distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials of a binary (yes/no) experiment. It is one of the most commonly used probability distributions in statistics.

Binomial distribution arises from a series of experiments known as Bernoulli trials. Each trial results in a success or a failure, and the probability of success is the same in each trial. The distribution is defined by two parameters:

The probability mass function of a binomial random variable is given by:

P(X = k) = \binom{n}k{}p^k(1 − p)^{n−k}

Where,

Properties of Binomial Distribution

Some of the key properties of Binomial Distribution are:

**Property **Formula
**Mean (Expected Value) μ = E(X) = np
**Variance σ2 = Var(X) = np(1 − p)
**Standard Deviation σ = √[np(1−p)​]
**Skewness Skewness = (1 - 2p)/[√[np(1−p)​]]
**Kurtosis Kurtosis = [1 − 6p(1 − p)]/[np(1 − p)]
**Probability Mass Function (PMF) P(X = k) = \binom{n}k{}p^k(1 − p)^{n−k}
**Moment Generating Function (MGF) MX​(t) = [pet + (1−p)]n

Examples of Binomial Distribution

Some of the examples of binomial distribution discussed as below:

What is Poisson Distribution?

**Poisson Distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event. It is named after the French mathematician Siméon Denis Poisson.

Random variable X follows a Poisson distribution if it represents the number of events occurring in a fixed interval of time or space. The probability mass function of a Poisson random variable is given by:

**P(X = k) = (λ k e −λ )/k!​

Where:

Properties of Poisson Distribution

Some of the key properties of Poisson Distribution are:

**Property **Formula
**Mean (Expected Value) μ = E(X) = λ
**Variance σ2 = Var(X) = λ
**Standard Deviation σ = √λ​
**Skewness Skewness = 1/√λ​
**Kurtosis Kurtosis = 1/λ​
**Moment Generating Function (MGF) M_{X}​(t) = e^{λ(e^t − 1)}

Examples of Poisson Distribution

Some of the most common examples which can be modelled using poison distribution are:

Binomial Vs. Poisson Distribution

Key differences between binomial and poison distribution are listed in the following table:

**Feature **Binomial Distribution **Poisson Distribution
**Definition Models the number of successes in a fixed number of independent trials, each with the same probability of success. Models the number of events occurring in a fixed interval of time or space, with events happening at a constant mean rate.
**Probability Mass Function (PMF) P(X = k) = \binom{n}k{}p^k(1 − p)^{n−k}Where _n is the number of trials and _p is the probability of success. **P(X = k) = (λ k e −λ )/k!​Where _λ is the average number of occurrences.
**Mean (Expected Value) _μ = _np μ = E(X) = λ
**Variance σ2 = Var(X) = np(1 − p) __σ_2 = _λ
**Standard Deviation σ = √[np(1−p)​] _σ = √__λ_​
**Skewness Skewness = (1 - 2p)/[√[np(1−p)​]] Skewness = 1/√λ​
**Kurtosis Kurtosis = [1 − 6p(1 − p)]/[np(1 − p)] Kurtosis = 1/λ​
**Moment Generating Function (MGF) MX​(t) = [pet + (1−p)]n M_{X}​(t) = e^{λ(e^t − 1)}
**Parameter Constraints n is a positive integer, 0 ≤ p ≤ 1 _λ > 0

Similarities Between Binomial and Poisson Distribution

Some of the common similarities between binomial and poison distribution are:

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