Why use poisson approximation

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Your Money. Personal Finance. Your Practice. Popular Courses. What Is a Poisson Distribution? Poisson distributions are used when the variable of interest is a discrete count variable. Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation. Teachers spend hours wading through derivations, equations, and theorems. A Poisson process is a model for a series of discrete events where the average time between events is known, but the exact timing of events is random. The arrival of an event is independent of the event before waiting time between events is memoryless.

All we know is the average time between failures. The failures are a Poisson process that looks like:. We know the average time between events, but the events are randomly spaced in time stochastic.

We might have back-to-back failures, but we could also go years between failures because the process is stochastic. The last point — events are not simultaneous — means we can think of each sub-interval in a Poisson process as a Bernoulli Trial, that is, either a success or a failure. Common examples of Poisson processes are customers calling a help center, visitors to a website, radioactive decay in atoms, photons arriving at a space telescope and movements in a stock price.

In the case of stock prices, we might know the average movements per day events per time , but we could also have a Poisson process for the number of trees in an acre events per area. One example of a Poisson process we often see is bus arrivals or trains. Jake VanderPlas has a great article on applying a Poisson process to bus arrival times which works better with made-up data than real-world data. We need the Poisson distribution to do interesting things like find the probability of a given number of events in a time period or find the probability of waiting some time until the next event.

The Poisson distribution probability mass function pmf gives the probability of observing k events in a time period given the length of the period and the average events per time:. With this substitution, the Poisson Distribution probability function now has one parameter:. We can think of lambda as the expected number of events in the interval.

The discrete nature of the Poisson distribution is why this is a probability mass function and not a density function. The graph below is the probability mass function of the Poisson distribution and shows the probability y-axis of a number of events x-axis occurring in one interval with different rate parameters. This makes sense because the rate parameter is the expected number of events in one interval. Therefore, the rate parameter represents the number of events with the greatest probability when the rate parameter is an integer.

When the rate parameter is not an integer, the highest probability number of events will be the nearest integer to the rate parameter. We can use the Poisson distribution pmf to find the probability of observing a number of events over an interval generated by a Poisson process.

We could continue with website failures to illustrate a problem solvable with a Poisson distribution, but I propose something grander. When I was a child, my father would sometimes take me into our yard to observe or try to observe meteor showers. In a typical meteor shower, we can expect five meteors per hour on average or one every 12 minutes. From these values, we get:. To test his prediction against the model, we can use the Poisson pmf distribution to find the probability of seeing exactly three meteors in one hour:.

If we went outside and observed for one hour every night for a week, then we could expect my dad to be right once! We can use other values in the equation to get the probability of different numbers of events and construct the pmf distribution. The graph below shows the probability mass function for the number of meteors in an hour with an average of 12 minutes between meteors, the rate parameter which is the same as saying five meteors expected in an hour.

The most likely number of meteors is five, the rate parameter of the distribution.