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:
- n: The number of trials.
- p: The probability of success in a single trial.
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,
- n is the number of trials,
- k is the number of successes,
- p is the probability of success,
\binom{n}{k} is the binomial coefficient, representing the number of ways to choose k successes out of n trials.
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) | |
| Moment Generating Function (MGF) | MX(t) = [pet + (1−p)]n |
Examples of Binomial Distribution
Some of the examples of binomial distribution discussed as below:
- Quality Control in Manufacturing: A factory produces light bulbs, and each bulb has a 2% probability of being defective. If a random sample of 100 light bulbs is taken, we can use the binomial distribution to find the probability of a certain number of defective bulbs.
- Clinical Trials: A new drug is being tested, and it is known to be effective in 70% of the cases. In a clinical trial, 10 patients are treated with this drug.
- Sports Statistics: In a basketball game, a player has a free throw success rate of 80%. During a game, the player takes 15 free throws.
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) = (λke−λ)/k!
Where:
- k is the number of occurrences,
- λ is the average number of occurrences in the interval,
- e is the base of the natural logarithm (approximately equal to 2.71828).
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) |
Examples of Poisson Distribution
Some of the most common examples which can be modelled using poison distribution are:
- Call Center: A call center receives an average of 5 calls per minute. We can use the Poisson distribution to find the probability of receiving a certain number of calls in a minute.
- Traffic Flow: On average, 2 cars pass through a checkpoint every 10 minutes. We can use the Poisson distribution to find the probability of a certain number of cars passing through the checkpoint in 10 minutes.
- Web Traffic: A website receives an average of 10 visits per hour. We can use the Poisson distribution to find the probability of receiving a certain number of visits in an hour.
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) | Where n is the number of trials and p is the probability of success. | P(X = k) = (λke−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) | σ2 = 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 | |
| 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:
- Both the binomial and Poisson distributions are discrete probability distributions, meaning they model the occurrence of discrete events.
- Both distributions take non-negative integer values (k = 0, 1, 2, . . .).
- Both distributions describe the probability of a certain number of events occurring in a given context (trials for binomial, time/space interval for Poisson).
- Both distributions assume independence in their events:
- Binomial distribution assumes independent trials.
- Poisson distribution assumes independent events occurring in a given interval.
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