Probability Distributions
2 Functions You Should Know Before Using Poisson Distribution
Where do we use the Poisson distribution?
The Poisson distribution depends on the number of independent random events which eventuate in a specific region or an interval. We can use it to find the probability of a particular event occurring a given number of times an interval. The term interval is usually time. For example, the probability of the number of x vehicles crossing a highway between 13:00 and 14:00. The probability of the number of x vehicles that park in a parking place in the specified time interval. These are the examples where we use the Poisson distribution to get the probability values.
According to these use cases of the Poisson distribution, it appears to be similar to the Exponential Distribution, but there is an important point that separates them from each other. The exponential distribution concerns the probability of the second that event occurs in an interval, unlike the Poisson distribution. It is the difference between the exponential distribution and Poisson distribution.
How is Poisson calculated?
There are two types of functions as mass and density function for getting probability values from distribution. Density functions are used for continuous probability distributions. Mass functions are used for discrete probability distributions. Since the Poisson distribution is a discrete probability distribution, we use the term probability mass function. So, how do we know the Poisson distribution is discrete. As mentioned earlier, Poisson finds the probability of the number of times a particular event occurs. So, the number of times can’t be 3.435 or 1.123, they can be an integer like 5, 1, 12. It means the values in the variable distributed as Poisson should be discrete.
Formula 1 shows the formula of the Poisson probability mass functions. As you can see, there are two parameters as “x” and “lambda”. Lambda represents the average number of events occurring over a given interval. Some sources can also present this average as “mu” because it is an average value. The “X” value represents the number of times that event will occur when the average number of events is equal to the lambda value.
Let’s try to solve an example probability problem to understand how probability values are calculated using Poisson’s probability mass function. Assume that we observed the Bosphorus bridge between 3 PM and 4 PM for ten days. We kept the number of vehicles that cross. The value of lambda should be the mean of the number of vehicles that cross from the bridge each day of 10. Suppose, the mean value is found to be 100. We want to obtain the probability of 80 vehicles crossing between 3 PM and 4 PM the next day. Then, the value of “x” should be 80. Let’s put the numbers where they belong.
As you can see from the resulting probability of 80 cars passing was found as 0.0051. It is close to zero that is a very low probability. It is because we find the likelihood of exact 80 vehicles crossing the bridge. Therefore, obtaining such a low probability is normal. Then, which value of “x” gives the highest likelihood? To understand the relation between “x” and “lambda” you can see Figure 1.
Figure 1 shows the difference between probabilities of the number of cars crossing based on three different lambda values (average number of cars crossing). The highest probability is obtained always when the number of vehicles is equal to the value of Lambda. Therefore, as the number of vehicles starts moving away from the value of lambda in both directions, the probabilities begin to decrease. However, as the lambda value increases, the average probability value decreases. It is because there are much more events that can occur in the interval.
Note: Looking at the shape of the distribution from Figure 1, the Poisson and normal distribution may appear to be the same. However, the variance in the Poisson distribution is equal to its mean, unlike the normal distribution. Besides, the normal distribution is a continuous probability distribution.
Cumulative Distribution Function
In the previous example, we’ve found the probability of exact 80 cars crossing from the bridge. It is also possible to find the likelihood of more than or less than 80 crossings from that bridge. It is what the cumulative distribution function (CDF) of poison distribution stands for. CDF takes the probability sum of each event up to the value “k”. You can see Formula 3 to understand what the CDF of Poisson distribution does.
Let’s find the probability of fewer than 80 cars passing over the bridge when the average number of passage is 100.
The probability value is 0.02264. It means that the number of vehicles crossing the bridge between 15:00 and 16:00 is unlikely to be less than 80. If we subtract 0.02264 from 1, the value 0.97736 gives the probability that more than 80 cars will cross the bridge between 15:00 and 16:00.
Conclusions
The examples for the Poisson distribution usually depend on time intervals. However, it is also possible to define your custom range. The range can also be by region. For example, the number of raindrops falling on 100 meters of the field or the number of vehicles parks the first 100 meters of the street. You can use the Poisson distribution to find the probability of the number of events, as long as you set a boundary that covers the occurrence of events.
I also highly recommend you to know the Exponential distribution to understand the difference between ED and PD. However, you might be interested in getting the probability of occurrence of the particular event at a specific second. If so, please see my other article on Exponential distribution.
