what is a poisson distribution 
in statistics a poisson distribution is a discrete probability distribution that tells how many times an event is likely to occur over a specified period it is a count distribution the parameter of which is lambda the mean number of events in the specific interval as the poisson distribution is a discrete function the variable can only take specific values in a potentially infinite list put differently the variable cannot take all values in any continuous range for the poisson distribution the variable can only take whole number values 0 1 2 3 etc with no fractions or decimals poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time it was named after french mathematician sim on denis poisson understanding poisson distributionsa poisson distribution can be used to estimate how likely it is that something will happen x number of times for example if the average number of people who buy cheeseburgers from a fast food chain on a friday night at a single restaurant location is 200 a poisson distribution can answer questions such as what is the probability that more than 300 people will buy burgers the application of the poisson distribution thereby enables managers to introduce optimal scheduling systems that would not work with say a normal distribution one of the most famous historical practical uses of the poisson distribution was estimating the annual number of prussian cavalry soldiers killed due to horse kicks modern examples include estimating the number of car crashes in a city of a given size in physiology this distribution is often used to calculate the probabilistic frequencies of different types of neurotransmitter secretions or if a video store averaged 400 customers every friday night what would have been the probability that 600 customers would come in on any given friday night formula for the poisson distributionf x xx e where e euler s number e 2 71828 x number of occurrencesx factorial of x equal to the expected value ev of x when that is also equal to its variance begin aligned f x frac lambda x x e lambda textbf where e text euler s number e 2 71828 dots x text number of occurrences x text factorial of x lambda text equal to the expected value ev of x text when that is also equal to its variance end aligned f x x x e where e euler s number e 2 71828 x number of occurrencesx factorial of x equal to the expected value ev of x when that is also equal to its variance given data that follows a poisson distribution it appears graphically as in the example depicted in the graph above assume that some operational process has an error rate of 3 if we further assume 100 random trials the poisson distribution describes the likelihood of getting a certain number of errors over some period of time such as a single day if the mean is very large then the poisson distribution is approximately a normal distribution the poisson distribution in financethe poisson distribution is also commonly used to model financial count data where the tally is small and is often zero as one example in finance it can be used to model the number of trades that a typical investor will make in a given day which can be 0 often 1 or 2 etc as another example this model can be used to predict the number of shocks to the market that will occur in a given time period say over a decade 