Poisson Distribution as an Approximation to Binomial Distribution

What is Poisson Distribution?

Poisson distribution describes the likelihood of a certain number of events occurring within a given time frame. It applies to situations where events happen independently and at a constant average rate. This distribution proves useful when numerous trials exist, each with a minimal chance of success. In layman’s language, it helps in forecasting how often something occurs over a specific time, provided we know the average rate of occurrence. Under Poisson Distribution,...

Probability Distribution Function (PDF) of Poisson Distribution

The Probability Density Function (PDF) of a Poisson distribution helps in determining the chance of a certain number of events happening within a set time or space. It is handy when events are infrequent and happen independently. For instance, imagine a bakery. Customers arrive randomly at an average rate of 4 customers per hour. We can apply the Poisson distribution to calculate the probability of a specific number of customers arriving in one hour. The PDF of Poisson Distribution can be written as:...

Characteristics of Poisson Distribution

The Poisson distribution is used for events that are rare, independent, and where the number of events in a given time or space is small. Given below are some important characteristics of the Poisson distribution:...

Shape of Poisson Distribution

A poison distribution is positively skewed. However, the degree of skewness decreases with an increase in the value of ....

Mean and Variance of Poisson Distribution

In the Poisson distribution, the mean and variance of the distribution are the same because this distribution is characterised by a property where the average rate of event occurrences (λ) is also equal to the spread or variability in the distribution. This property is specific to the Poisson distribution and is a key feature of it. Therefore, The expected mean E(X) and the variance V(X) are both equal to λ....

Fitting a Poisson Distribution

Fitting a Poisson Distribution involves finding the best match between observed data and the Poisson model. It is like searching for a hat that fits just right. Suppose, there is data on how often something happens, like the number of customers arriving at a store each hour. If this data follows a pattern where events are rare and occur independently, the Poisson distribution might be a good fit. To fit it, one compares the actual data to what the Poisson model predicts. If they match up well, it suggests that the Poisson distribution accurately describes the situation. In practical terms, fitting a Poisson Distribution helps in understanding and making predictions about situations involving rare events, like customer arrivals or machine failures. It is a bit like finding the right puzzle piece that fits snugly into the data available....

Poisson Distribution as an Approximation to Binomial Distribution

The Poisson distribution simplifies the Binomial distribution. The Binomial distribution deals with the number of successes in a set of trials. However, when there are many trials with a small chance of success, it becomes impractical to calculate using the Binomial probabilities. In such situations, the Poisson distribution works as a shortcut. It focuses on the frequency of events in a specific span of time or space, especially for rare occurrences....

Examples of Poisson Distribution

Example 1:...