The Poisson Distribution models the probability of a given number of events occurring in a fixed interval of time or space, provided the events happen independently and at a constant average rate. It is a discrete probability distribution.
Mean and Variance:
\[ \mu = \lambda, \quad \sigma^2 = \lambda \]
Probability Mass Function (PMF):
\[ f(x) = \frac{\lambda^x e^{-\lambda}}{x!} \]
Cumulative Distribution Function (CDF):
\[ F(x_j) = \sum_{k \leq j} \frac{\lambda^k e^{-\lambda}}{k!} \]