When is poisson distribution applicable

How long can we expect to wait to see the next meteor if we arrive at a random time? My dad always this time optimistically claimed we only had to wait six minutes for the first meteor, which agrees with our intuition.

The probability of waiting a given amount of time between successive events decreases exponentially as time increases. The following equation shows the probability of waiting more than a specified time. With our example, we have one event per 12 minutes, and if we plug in the numbers, we get a We can expect to wait more than 30 minutes, about 8. Note this is the time between each successive pair of events.

The waiting times between events are memoryless, so the time between two events has no effect on the time between any other events. This memorylessness is also known as the Markov property. There is a percent chance of waiting more than zero minutes, which drops off to a near-zero percent chance of waiting more than 80 minutes. Rearranging the equation, we can use it to find the probability of waiting less than or equal to a time:.

We can expect to wait six minutes or less to see a meteor To visualize the distribution of waiting times, we can once again run a simulated experiment. We simulate watching for , minutes with an average rate of one meteor per 12 minutes.

Then we find the waiting time between each meteor we see and plot the distribution. The graph below shows the distribution of the average waiting time between meteors from these trials:. The average of the 10, runs is Surprisingly, this average is also the average waiting time to see the first meteor if we arrive at a random time.

At first, this may seem counterintuitive: if events occur on average every 12 minutes, then why do we have to wait the entire 12 minutes before seeing one event? The answer is we are calculating an average waiting time, taking into account all possible situations.

However, because waiting time is an exponential distribution, sometimes we show up and have to wait an hour, which outweighs the more frequent times when we wait fewer than 12 minutes. The average time to see the first meteor averaged over all the occurrences will be the same as the average time between events. The average first event waiting time in a Poisson process is known as the Waiting Time Paradox. Well, this time we got precisely the result we expected: five meteors. We had to wait 15 minutes for the first one then 12 minutes for the next.

The next time you find yourself losing focus in statistics, you have my permission to stop paying attention to the teacher. Applying technical concepts helps you learn the material and better appreciate how stats help us understand the world. Above all, stay curious: There are many amazing phenomena in the world, and data science is an excellent tool for exploring them.

This article was originally published on Towards Data Science. Statistics can be fun if you learn key concepts the right way. Will Koehrsen. Obviously the variance will be larger in the second case. You are probably most familiar with the normal distribution, because it underlies most of the standard statistical procedures that we use. For the normal distribution the mean and variance are independent, and there we would not expect the variance to increase as the mean does.

An important, though unfortunate, feature of many samples of data is that the variability of the results is greater than would be predicted by the Poisson distribution. The example used here is probably a good example of what can go wrong. You should recall that I assumed at the beginning that day to day observations of the number of calls are independent of one another. Thus, for example, the fact that we had 5 calls today should not be relevant in predicting the number of calls we will receive tomorrow.

However, if we are dealing with sexual harassment, I would think it likely that observations are not truly independent.

There is probably some seasonable variation in harassing behaviors. It seems reasonable, for example, that women would receive fewer obnoxious remarks when they wear bulky sweaters in the winter than they would when they wear lighter clothing in the summer. If this were the case, the variability of the daily frequencies would reflect not only the natural variability we expect with a Poisson distribution, but also variability due to seasonal causes.

Thus the actual variance is likely to exceed m. The result of having overdispersion is that the Poisson distribution may not completely model the data at hand.

There really is very little that we can do about this, unless we can find a model for the increased variance, but it is important to recognize. We find that the Poisson is a very nice model for many kinds of data, but don't expect that it will model everything. The reason why I have discussed the Poisson distribution is that it is frequently a useful way of modeling categorical data.

This is particularly important when the overall sample size N is not fixed, but is treated as a random variable. We can model each category count as a Poisson variable, and derive our hypothesis tests, and confidence intervals, on the basis of that model.

Thus we might take each of the four cell counts in a 2X2 contingency table as an independent Poisson variable. Given this statistic, we might be interested in asking about the probability that an elementary school teacher will have at least one child in his classroom who suffers from Tourette's syndrome.

Notice that this problem is a bit different from the one we discussed with the Poisson distribution. In that situation I new the mean number of complaints per day of sexual harassment, and was interested in asking about the probability of receiving no calls today or any other value that I might wish. But when I am faced with the example of Tourette's syndrome, it is logical for me to ask about the size of the class. For instance, I could ask "Out of a class of 20 students, how likely is it that one student will suffer from Tourette's syndrome?

For this situation I am going to fall back on the binomial distribution. This is a distribution that asks about the probability of x events out of N events. In other words, it allows me to ask about the probability of 1 or 2, or 3, etc. Tourette's child out of a class of Suppose that the teacher has 20 children in his class and he wants to know that probability that 1 of them will be a Tourette's child.

The probability can be obtained as. Thus the probability is. Perhaps our teacher is more interested in knowing the probability that none of this children would have Tourette's syndrome. Select basic ads. Create a personalised ads profile. Select personalised ads. Apply market research to generate audience insights.

Measure content performance. Develop and improve products. List of Partners vendors. In statistics , a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time.

The Poisson distribution is a discrete function, meaning that 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 a discrete distribution , the variable can only take the values 0, 1, 2, 3, etc. A 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 , a Poisson distribution can answer questions such as, "What is the probability that more than people will buy burgers?

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 customers every Friday night, what would have been the probability that customers would come in on any given Friday night?

Given data that follows a Poisson distribution, it appears graphically as:.