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Poisson Distribution

A Poisson distribution is a mathematical technique used to show the number of times a given event is likely to happen within a specified period. It is a discrete function, which means, an event can only be displayed either as occurring or not occurring and this means the variables being observed can only be displayed in whole numbers. Poisson distribution is used to measure independent events that happen at a constant rate within a specified duration of time.

Table Of Contents
  • When is Poisson distribution valid?
  • The difference between a Poisson distribution and a normal distribution
  • How Poisson distribution is used in real life

When is Poisson distribution valid?

The Poisson distribution will be considered a viable probability analysis tool only if the event being analyzed meets the following conditions:
  • The event occurs k times within a given period and the potential values for k are whole numbers like 0, 1, 2, 3, etc.
  • One occurrence of the event does not affect the likelihood of the event occurring again
  • The event being observed does not occur multiple times at one given time. There has to be some time interval to separate the occurrences of the event (even if it’s just one second).
  • The likelihood of the event occurring within a section of the time frame being analyzed is proportional to the duration of that small section of the timeframe.
  • The chances of the event happening (number of trials) are significantly higher than the defined number of times that the event actually occurs.

The difference between a Poisson distribution and a normal distribution

The normal distribution is so common in mathematics and statistics that most people tend to forget that it is not always so ubiquitous in the actual data. And since it is continuous, most people classify all numerical variables as continuous but in the actual sense, numerical variables can take a continuous or a discrete form. The easiest way to tell the difference between continuous and discrete variables is to know that discrete variables only take whole numbers and continuous variables can be any number. For instance, 4.973 is a continuous variable and 5 is a discrete variable. Normal distributions are continuous while Poisson distributions are discrete.

How Poisson distribution is used in real life

To understand the application of Poisson distribution, let's consider the following example:
A coffee shop sells 200 cups of coffee every Sunday morning. Using this information, one can predict the likelihood that more coffee will be sold (perhaps 300 or 400 cups) on the following Sunday mornings. Another example we could consider is the number of customers who visit a store every day. If in seven days the number averages to 1, 000, then one can predict the likelihood of some days having more customers than others.
Because of this ability to effectively tell the likelihood of events happening, Poisson distribution is used by businesses and companies to predict the number of sales or customers on certain days of the week or seasons of the year. When it comes to a business setting, overstocking may sometimes bring losses if the products are not sold at the right time. Similarly, having fewer products in stock could still bring losses because the business is not maximizing its sales. Using Poisson distribution enables businesses to make estimates of the times there is a higher demand so that they can stock more products. Restaurants and hotels, for instance, could prepare for the rise in the number of customers by hiring extra workers well in advance, buying more supplies, or having contingency plans in place just in case they are not able to accommodate more guests. Poisson distribution generally helps companies adjust to supply and demand effectively, which in turn keeps the businesses bringing in more sales. It also prevents waste of resources.