# Poisson Distribution

**Episode #5 of the course Theory of probability by Polina Durneva**

Good morning!

Today, we will talk about the Poisson distribution that is mainly used to predict the probability of a given number of events happening over a period of time. This distribution is closely associated with other types of probability distributions, such as the negative exponential distribution, that we will discuss later in the course. So, let’s get started!

**What Is Poisson Distribution?**

In the previous distributions that we discussed, an event could be classified as either success or failure. This can also be applicable to the Poisson distribution. It means that we want to know the probability that a number of success occurs over a fixed period of time. For example, you own a restaurant where 100 people come to on average every day. You might want to know the probability that more than 100 people will come to your restaurant today. To calculate such probability, you will use the Poisson distribution! A hundred people will be 100 of success, and a day will be a fixed period of time.

Before we proceed to actual calculations of the probability, let’s discuss the main assumptions held for the Poisson distribution:

1. The intervals over which the events occur do not overlap.

2. The events are independent.

3. The probability that more than one event happens in a very short time period is approximately 0.

**Formula for Poisson Distribution**

It takes a while to derive the formula for the Poisson distribution that takes the following form: Probability (x) = (λ^{x}e^{-λ})/x!, where x is the Poisson random variable and λ is the expected number of events (or the average).

Let’s use our previous example with the restaurant to calculate the probability that 125 customers will come to our restaurant today. As we know, on average, there are 100 customers in our restaurant every day (this is λ). Then, the probability will be: P(x = 125) = (λ^{x}e^{-λ})/x! = (100^{125}e^{-100})/125! ≈ 0.00198 = 0.198%. This shows that the probability that we will have 125 customers today is less than 1%.

**Expected Value of Poisson Distribution**

As stated previously, λ is the expected value for the Poisson distribution. It means that the average value would be the rate of occurrence of an event. In our previous example, the expected value was 100 customers per day.

**Variance of Poisson Distribution**

The variance of this type of distribution is also easy to remember because it equals to the expected value. In the example we used, the variance would also be 100.

That’s it for today! In the next three days, we will discuss discrete probability topics such as the uniform distribution (by the way, the uniform distribution can also be continuous, but we will not talk about this case in this course), the geometric distribution, and the negative binomial distribution. The rest of the course will be devoted to the continuous distributions.

Take care,

Polina

**Recommended book**

*Lady Luck: The Theory of Probability* by Warren Weaver

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