September 17

Poisson r.v.

The Poisson random variable, denoted by $X$, models the number of occurrences of an event within a fixed interval of time or space, given that the events occur independently and at a constant average rate. The Poisson distribution is often used for rare events.

• The Poisson distribution is discrete, so it only takes non-negative integer values.
• It is parameterized by a single parameter $\lambda$, which represents the average rate of occurrence of the event.
• The distribution is commonly used in modeling counts of events like the number of phone calls received at a call center in a given hour, or the number of accidents occurring at an intersection in a month.

The Poisson distribution is the limiting case of the Binomial distribution as $n \to \infty$ and $p \to 0$, such that $\lambda = np$ (where $\lambda$ is the average rate of occurrence). The probability mass function (PMF) of a Poisson random variable is:

\[P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}, \quad x = 0, 1, 2, \dots\] \[E(X) = \lambda\] \[Var(X) = \lambda\]

Where $\lambda$ is the rate of occurrence (the expected number of events in the interval), and $e$ is Euler’s number (approximately 2.71828).

Example: Website Server Requests

Scenario: A web server receives an average of 3 requests per minute. We want to model the number of requests received in a minute using a Poisson distribution.

• Parameter: The average rate of occurrence is given, so $\lambda = 3$.

• Question 1: What is the probability that the server receives exactly 5 requests in the next minute?
  • Using the PMF with $x = 5$ and $\lambda = 3$: \(P(X = 5) = \frac{3^5 e^{-3}}{5!} = \frac{243 \times 0.0498}{120} \approx 0.1008\)
  • The probability is approximately 10.08%.
• Question 2: What is the expected number of requests and the variance of the number of requests in a minute?
  • The expected number of requests is the mean, which is $\lambda$. \(E(X) = 3\)
  • The variance is also equal to $\lambda$. \(Var(X) = 3\)