Understanding Poisson Distribution.

In today’s blog post we cover the following:

• Recall what probability distribution is❓

• Expected Value.

• Poisson distribution.

• Real world Examples.

Before jumping onto Poisson distribution let’s quickly recall what a probability distribution is:

Let 𝑍 be some random variable. The probability distribution function of 𝑍 assigns probabilities to the different outcomes 𝑍 can take. Graphically, probability of an outcome is proportional to the height of the curve.

We can divide random variables into three classifications:

• 𝑍 is discrete: Discrete random variables may only assume values on a specified list. Things like populations, movie ratings etc. are all discrete random variables.

• 𝑍 is continuous: Continuous random variable can take on arbitrarily exact values. For example, temperature, speed, time etc. are all continuous random variables.

• 𝑍 is mixed: Mixed random variables assign probabilities to both discrete and continuous random variables, i.e. it is a combination of the above two categories.

Another important thing to know before we jump in is Expected value.

• Expected value (EV) is one of the most important concepts in probability. The EV for a given probability distribution can be described as “the mean value for many repeated samples(random variables) from that distribution.

Poisson Distribution:

• Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

• For instance, a call centre receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive.

• We say 𝑍 is Poisson-distributed if:

Understanding 𝑘 and 𝜆:

• 𝜆 is called a parameter of the distribution, and it controls the distribution’s shape. For the Poisson distribution, 𝜆 can be any positive number. By increasing 𝜆, we add more probability to larger values, and conversely by decreasing 𝜆 we add more probability to smaller values. One can describe 𝜆 as the intensity of the Poisson distribution.

• Unlike 𝜆 , which can be any positive number, the value 𝑘 in the above formula must be a non-negative integer, i.e., 𝑘 must take on values 0,1,2, and so on.

• This is very important, because if you wanted to model a population you could not make sense of populations with 4.25 or 5.612 members.

One useful property of the Poisson distribution is that its expected value is equal to its parameter:

• 𝐸[𝑍|𝜆]=𝜆

• Now I believe it is more clear why increasing 𝜆 assigns more probability to lager values and vice-versa

Let’s make our understanding concrete with the following real world example:

• The average number of goals in a FIFA World Cup match is approximately 2.5 and the Poisson model is appropriate. Because the average event rate is 2.5 goals per match, λ = 2.5.

In the above example I used the phrase assuming the Poisson model is correct, now the question arises when does this model fail or the assumption can not be made. Let’s see another example:

• The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).

• The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.

Rule of thumb!!

• So before assuming the Poisson model to be true ask yourself two question whether the rate of occurrence of events is somewhat constant or not and if the events are occurring independent of each other.

Hope you enjoyed reading this.
Cheers!! 🍻

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