Poisson Distribution in Python – understand its background and calculation method
In Python, the Poisson Distribution (Poisson Distribution) is a very commonly used probability distribution, especially when dealing with discrete random variables. The Poisson distribution is used to estimate the probability of an event occurring over a certain period of time. In this article, we’ll explore the background and computation of the Poisson distribution, and how it can be used in Python to solve real-world problems.
Background on the Poisson distribution
The Poisson distribution was invented by the French mathematician Simone Denis Poisson in 1837 to describe the probability distribution of an event occurring within a certain period of time. The Poisson distribution is suitable for those discrete random variables that satisfy the following conditions:
- The number of events that occurred in a certain period of time
- The probability of an event occurring does not change at any moment
- events are independent
The Poisson distribution has been widely used in many fields, such as:
- Distribution of Explosive Quantities
- cosmic ray landing frequency
- Distribution of network traffic
Calculation method of Poisson distribution
The probability mass function (PMF) equation for the Poisson distribution is as follows:
P
(
x
=
k
)
=
e
?
lambda
lambda
k
k
!
P(X = k) = e^{-\lambda}\frac{\lambda^{k}}{k!}
P(X=k)=e?λk!λk?
in:
-
x
x
X is a random variable representing the number of times an event occurs over a period of time
-
lambda
\lambda
λ represents the average occurrence rate of events during this period
-
e
e
e is the base of the natural logarithm (ie
2.718281828
\mathit{2.718281828}
2.718281828)
-
k
k
k is a non-negative integer representing the actual number of events that occurred during this time
Using Python code, we can calculate the probability mass function of the Poisson distribution. Here is a simple code example:
from math import exp
from math import factorial
def poisson_probability(k, lmbda):
return exp(-lmbda) * (lmbda ** k) / factorial(k)
The function accepts two parameters: the actual number of event occurrences (k) and the average occurrence rate of the event (
lambda
\lambda
lambda). This function uses the exp and factorial functions from the math module to compute the probability mass function of the Poisson distribution.
Using the Poisson distribution in Python
Now, let’s look at some practical examples using the Poisson distribution. In these examples, we’ll use traffic data from the University of Michigan’s School of Electrical and Computer Engineering.
Example 1: Calculate traffic during a specific time period
Let’s say we want to monitor website traffic for an hour. We can use the Poisson distribution to calculate the probability of having 50 visits during this hour. Please see the following code:
from math import exp
from math import factorial
def poisson_probability(k, lmbda):
return exp(-lmbda) * (lmbda ** k) / factorial(k)
# traffic data
lambda_hour = 45
# Calculate the probability that the number of visits is 50 in a certain period of time
k = 50
prob_50 = poisson_probability(k, lambda_hour)
print(f"The probability of having {<!-- -->k} visits in one hour is {<!-- -->prob_50:.4f}")
According to the output of this code, the probability of having 50 visits is about 0.0257.
Example 2: Calculate the traffic difference between two time periods
Now, we will use the Poisson distribution to calculate the difference in traffic to the website in the two time periods. Suppose we have two hours of traffic data, the first hour’s
lambda
1
\lambda_1
λ1? is 45, the second hour
lambda
2
\lambda_2
λ2? is 55. We want to know what is the probability that the number of visits in the second hour is higher than in the first hour. Please see the following code:
from math import exp
from math import factorial
def poisson_probability(k, lmbda):
return exp(-lmbda) * (lmbda ** k) / factorial(k)
# traffic data
lambda_hour1 = 45
lambda_hour2 = 55
# Calculate the probability that the number of visits in the second hour is higher than that in the first hour
prob_diff = sum(poisson_probability(k, lambda_hour2-lambda_hour1)
for k in range(0, 50))
print(f"The probability of having more visits in hour 2 than hour 1 is {<!-- -->prob_diff:.4f}")
According to the output of this code, the probability that the number of visits in the second hour is higher than that in the first hour is about 0.8022.
Conclusion
In Python, the Poisson distribution is a very useful probability distribution, especially when dealing with discrete random variables. Through this article, we understand the background of the Poisson distribution, learn the calculation method of the Poisson distribution, and see some practical examples, including calculating the flow rate in a specific period of time and calculating the difference in flow rate between two periods of time. If you are interested in Python and probability distributions, I recommend you to continue learning and exploring them, you will find them very useful for fields such as data analysis, machine learning and artificial intelligence.
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