Poisson Distribution: Definition, Properties and applications

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The Poisson distribution is a powerful statistical tool used to model the probability of a certain number of events occurring within a fixed interval of time or space. Whether you’re trying to predict the number of customer service calls received per hour, the number of typos on a page, or the number of cars passing a certain point on a highway in a given minute, the Poisson distribution can provide valuable insights. This blog post will delve into the intricacies of the Poisson distribution, exploring its definition, properties, applications, and limitations.

What is the Poisson Distribution?

The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period. This means it deals with probabilities for a finite or countably infinite number of outcomes. Unlike continuous distributions, which can take on any value within a range, the Poisson distribution only considers whole numbers representing the count of events.

Formally, a discrete random variable X is said to have Poisson distribution if its probability function is defined as,

$f\left ( x;\lambda \right )=\frac{e^{-\lambda }\lambda ^{x}}{x!}; x=0,1,2,...,\infty$

where λ is the parameter of the distribution and it is the mean number of success. Also е=2.71828.

Mean of the distribution is E[x]= λ and Variance is Var[X]= λ. The mean and variance of the Distribution is equal.

Understanding the Rate Parameter (λ)

The rate parameter, λ, is the most crucial element of the Poisson distribution. It directly influences the shape and probabilities of the distribution. It represents the average number of events we expect to see within the given interval.

High λ: A large value of λ indicates a higher average rate of events. The distribution will be skewed less and centered further to the right, suggesting a higher probability of observing a larger number of events.
Low λ: A small value of λ indicates a lower average rate of events. The distribution will be highly skewed to the right, with a higher probability of observing a small number of events (including zero).

Properties of Poisson Distribution

Following properties are exist,

Poisson distribution has only one parameter named “λ”.
Mean of poisson distribution is λ.
It is only a distribution which variance is also λ.
Moment generating function is ${displaystyle exp[lambda (e^{t}-1)]}$ .
The distribution is positively skewed and leptokurtic.
It tends to normal distribution if λ⟶∞.

Derivation of Poisson Distribution from Binomial Distribution

Under following condition , we can derive the distribution from binomial distribution,

The probability of success or failure in bernoulli trial is very small that means which tends to zero. p⟶0 and q⟶0.
The number of trial n is very large that is n⟶∞.
λ=np is finite constant that is average number of success is finite.

we can also derive the it from poisson process.

Applications of Poisson Distribution

The Poisson distribution finds applications in a wide range of fields. Here are a few examples:

Telecommunications: Modeling the number of phone calls arriving at a call center per hour.
Traffic Analysis: Predicting the number of cars passing a specific point on a highway in a given minute.
Healthcare: Determining the number of patients arriving at an emergency room per day.
Manufacturing: Analyzing the number of defects found in a batch of products.
Insurance: Calculating the number of insurance claims filed per month.
Ecology: Estimating the number of trees of a certain species found in a given area.
Astronomy: Modeling the number of meteorites larger than a certain size that strike the Earth per year.
Queueing Theory: Analyzing the number of customers arriving at a service counter per unit of time.

Calculating Probabilities with the Poisson Distribution

To calculate probabilities using the Poisson distribution, you can use the PMF formula mentioned earlier, statistical software packages, or online calculators.

Let’s consider an example: Suppose a website receives an average of 5 hits per minute. What is the probability that the website receives exactly 3 hits in a given minute?

Here, λ = 5 and k = 3. Using the PMF formula:

P(X = 3) = (e<sup>-5</sup> * 5<sup>3</sup>) / 3!
= (0.006738 * 125) / 6
= 0.14037

Therefore, the probability of receiving exactly 3 hits in a given minute is approximately 0.14037, or 14.037%.

You can also calculate other probabilities, such as:

P(X < k): Probability of observing less than k events.
P(X > k): Probability of observing more than k events.
P(X ≤ k): Probability of observing at most k events.
P(X ≥ k): Probability of observing at least k events.

These probabilities can be calculated by summing the individual probabilities from the PMF formula or by using cumulative distribution functions (CDFs) available in statistical software.

Relationship to Other Distributions

Binomial Distribution: The Poisson distribution can be seen as a limiting case of the binomial distribution when the number of trials (n) is large and the probability of success (p) is small, such that n * p* approaches a constant value (λ). In other words, when you have many independent trials but each trial has a very small probability of success, the number of successes can be approximated by a Poisson distribution.
Exponential Distribution: The exponential distribution is related to the Poisson distribution. If the number of events follows a Poisson distribution, then the time between successive events follows an exponential distribution. The exponential distribution models the time until the next event occurs, while the Poisson distribution models the number of events occurring within a fixed interval.

Limitations of the Poisson Distribution

While a powerful tool, the Poisson distribution has its limitations:

Violation of Assumptions: If the assumptions of independence, randomness, and constant rate are violated, the Poisson distribution may not accurately model the data. For example, if events are clustered or if the rate of events changes significantly over time, a different distribution might be more appropriate.
Overdispersion: Overdispersion occurs when the variance of the data is greater than the mean. This can happen when there is unobserved heterogeneity or clustering in the data. In such cases, a negative binomial distribution might be a better alternative.
Underdispersion: Underdispersion occurs when the variance of the data is less than the mean. This is less common but can indicate that the events are more regularly spaced than expected under a Poisson process.
Zero-Inflated Data: If the data contains an excess of zero counts compared to what would be expected under a Poisson distribution, a zero-inflated Poisson (ZIP) model might be more suitable. This can occur when there is an additional process that generates zero counts.

Alternatives to the Poisson Distribution

When the assumptions of the Poisson distribution are not met, several alternative distributions can be considered:

Negative Binomial Distribution: This distribution is useful when dealing with overdispersed count data. It allows for the variance to be greater than the mean.
Zero-Inflated Poisson (ZIP) Distribution: This distribution is used when there are more zero counts than expected under a standard Poisson distribution.
Zero-Truncated Poisson Distribution: This distribution is used when zero counts are not possible or not observed in the data.
Gamma Distribution: When modeling the time between events (instead of the number of events), the Gamma distribution can be used if the rate parameter is not constant.
Quasi-Poisson Regression: This approach allows for the estimation of a dispersion parameter to account for over- or under-dispersion without changing the underlying distribution family.

Conclusion

The Poisson distribution is a versatile and widely used statistical tool for modeling the number of events occurring within a fixed interval. By understanding its assumptions, properties, and limitations, you can effectively apply it to a variety of real-world problems. When the assumptions are violated, it’s important to consider alternative distributions that better capture the characteristics of your data. By mastering the Poisson distribution and its related concepts, you gain a valuable tool for analyzing and interpreting count data in numerous fields. Data Science Blog