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# Poisson Discrete Distribution

Poisson Discrete Distribution is a statistical distribution that is used to model the number of events occurring in a fixed interval of time or space. The distribution is named after the French mathematician Simeon-Denis Poisson.

A random variable X is said to have a Poisson distribution with parameter λ (lambda) if its probability mass function (PMF) is given by:

P(X = k) = (e^-λ) * (λ^k) / k!

where k is a non-negative integer representing the number of events in the interval, and e is the mathematical constant approximately equal to 2.71828. λ is the average number of events that occur in the interval.

The Poisson distribution has the following properties:

• It is a discrete distribution, meaning that the possible values of X are non-negative integers.

• It is a non-negative distribution, meaning that the PMF is always non-negative.

• It is a memoryless distribution, meaning that the future events are not affected by the past events.

• The mean and variance of the distribution are equal to λ.

The Poisson distribution is widely used in various fields such as queuing theory, reliability engineering, and finance to model the number of occurrences of rare events. Examples of events that can be modeled using the Poisson distribution are:

• The number of calls received by a call center in a given time interval

• The number of customers arriving at a bank in a given time interval

• The number of goals scored by a soccer team in a given time interval

It is important to note that the Poisson distribution is only appropriate when the events are rare, independent, and occur with a constant rate in the interval.

Here's an implementation of the Poisson distribution in Python:

import math

def poisson_discrete_distribution(lambda_, k):
return math.exp(-lambda_) * (lambda_**k) / math.factorial(k)

lambda_ = 5
k = 3

result = poisson_discrete_distribution(lambda_, k)

print(result)

In this implementation, the poisson_discrete_distribution function calculates the probability of observing k events given the average rate of events lambda_. The math.exp function calculates the exponential of a given number, and the math.factorial function calculates the factorial of a given number.

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