Poisson Distribution: Calculating Event Probabilities

Poisson distribution. Poisson distribution allows you to calculate the probability of a certain number of events occurring in a certain time interval. Poisson distribution describes the distribution of binary data from an infinite sample.

Thus, it gives the probability of getting X events in a population. It sounds very similar to binomial probability, but there are some slight differences. Now, this is named, this particular distribution is named after Baron Simeon Denis Poisson.

He was born on June 21st, 1781 and passed on April 25th, 1840. He was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity, magnetism, and thermodynamics. Quite a busy guy.

You probably want to know what are the variables or parameters that make up the Poisson distribution. Well, you have X number of events, which is very similar to trials in the binomial probability function. You have the expected numeric value, and that is the mean.

So, that is whatever the average value is, and then you have the option to choose whether or not you want a cumulative calculation, which measures from a certain point and lower, or you're looking for an exact point on the line. So, that's the choice between either probability mass function, which represents a point on the line, which we can see here in the chart, is that little red point, or cumulative distribution, which represents the shaded area here. In this case, we're looking at values from zero down to negative, looks like seven or eight.

When should you use Poisson distribution? Well, given the average probability of an event, that's the mean, find the probability of a certain number of events happening, and this should take place within a specific time interval. So, let's take a look at some examples. We have some very simple examples here, and there are some exercises that you can do on your own.

On average, 30 employees are hired by the company each month. What is the probability that there will be exactly 27 new hires this month? The population represents all the employees, everyone. So, to calculate this, we're going to use Poisson distribution, which is the Excel function, Poisson distribution, and so we have a couple of values to select, less than binomial probability.

So, the first thing is X. X, in this case, is 27. That's the number we're trying to figure out the probability for. Then, we want to select the mean, which is 30.

That's the average number of people who get hired. And then, I choose either I, whether I want a probability mass function, which is a specific point, or cumulative distribution, where I select a certain value and then everything underneath it, or less than that. In this case, I want false.

So, I'll type false. I'll press ENTER. There's a 6.5 percent that exactly 27 new hires will occur this month.

That's the time period. So, that's the first exercise. The second one is what is the probability that the number of new hires will be 27 or less? So, here, we're looking for a range of values.

We're not just looking for 27, but we're also looking for 26,25,24,23, and so on, all the way down to zero, and beyond, if necessary. So, equal, post on. Then, again, I want to select X, which is 27.

I'll enter a comma. I'll select mean. I'll enter a comma, and here's the main difference.

This time, I'm going to choose true. So, I'm looking for 27 or less. So, I'm going to type true.

I'll press ENTER. 33 percent. So, there's a 33 percent that anywhere from 1 to 27 new hires will get hired this month.

So, it's not just 27, but any number in between 1 and 27. The next calculation is looking for a certain range. What if we didn't necessarily want to look for values that were 27 or less, but maybe between, let's say, 20 and 27? So, for this next exercise, we're looking at a range between 27 and 33.

So, what is the probability that the number of new hires this month will be between 27 and 33? All right. So, now, there isn't a specific function that allows us to enter 27 and 33. So, we have to enter 27 and 33 independently.

You want to start with the higher number, because you're going to subtract the lower number. So, there's going to be a lot of typing here. So, let's get started.

Equal, Poisson, distribution. Now, my starting number is 33. I'm going to enter a comma.

I'm going to select the mean, which is 30. That's the average number of people hired. Now, cumulative distribution is going to be true, because I want to grab everything from 33 on down.

Then I'm going to subtract that by Poisson distribution, and I'm going to select 26. I'll enter a comma. Again, I'm going to select the same mean.

I'll enter a comma, then I'll type true. I'll press TAB, close parentheses. Now, before I continue, you'll notice that we're looking for values between 27 and 33, but I wrote 26.

Why did I write 26 instead of 27? Well, the difference between 33 and 27 is 6. But I would like to also include 27. So, I'm going to move down one number, so 27 is included. That'll be the seventh number in the seven number range.

This is what's actually going to give me the correct calculation. Now, I'm going to press ENTER, and my calculation is 47.71. I can change the formatting to percentage, so it looks just like the answer I have below. I could do that for the prior answer as well.

So, that's how you would perform the calculation to measure the number of new hires between 27 and 33. Now, 47% is pretty high, and that's appropriate, because if your average is 30, you're looking at, like, potentially one standard deviation to the left or the right of the mean. Now, there is some exercises that you can do on your own that basically follow the same pattern as the last exercise.

I'll do one exercise that captures the difference between 8 and 12 employees using the same type of Poisson distribution. Actually, I should let you know that on this sheet, the answers are already present. I just have to change the font color to black, and there is the calculation.

And so, you can see here what you would write in the cell for your result. The answer is going to be, I'll change this back to white, and the answer is going to be 57.13. Now, the reason I'm going over this with you is, unlike the last exercise, we actually have values for the exercise for you as students, and we've plotted the information on a graph, and we used Poisson distribution for each point on the line. And this is the percentages that we returned.

If I select between 8 and 12, and I look at the percentages down at the bottom, it's 57.13, which is exactly the same as the result I get when I use the number 7 to 12. So, notice here, 8 to 12, which is the range that I was looking for, gave me a result, a probability of 57.13. But in my calculation, I go up to 7, and that actually captures 8 to 12. So, that's proof that going back one number does make a difference and is the appropriate thing to do.

So, in this section, we looked at Poisson distribution. Poisson distribution helps you calculate the probability of a certain number of events happening within a certain time frame. Thank you for watching.