The Poisson distribution is a fundamental concept in probability theory and statistics, used to model the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. Here's a breakdown of its key aspects:

Key Characteristics:

• Discrete Distribution:
  • The Poisson distribution deals with countable events, meaning the outcomes are whole numbers (0, 1, 2, 3, ...).
• Single Parameter (λ):
  • The distribution is characterized by a single parameter, λ (lambda), which represents the average number of events in the given interval.
  • Importantly, in a Poisson distribution, the mean and variance are both equal to λ.
• Independence:
  • The occurrence of one event does not affect the probability of another event.
• Constant Rate:
  • The average rate of events remains constant within the specified interval.

Formula:

The probability of observing k events in the given interval is given by the Poisson probability mass function (PMF):

$$ \operatorname{Pr}(X=k)=\frac{\lambda^k e^{-\lambda}}{k!} $$

Where:

• $P(X = k)$ is the probability of k events occurring.
• $e$ is Euler's number (approximately 2.71828).
• $λ$ is the average number of events (the rate parameter).
• $k$ is the number of events (a non-negative integer).
• $k!$ is the factorial of k.

Applications:

The Poisson distribution is widely used in various fields, including:

• Telecommunications:
  • Modeling the number of phone calls received by a call center in a given time period.
• Healthcare:
  • Analyzing the number of patient arrivals at an emergency room.
  • Studying the occurrence of rare diseases.
• Finance:
  • Modeling the number of stock market crashes in a given period.