The Poisson Distribution is typically used to model the number of events that happen in a certain time period, especially where the expected number of events is very small. The Poisson distribution is parameterized simply by $\lambda$, which can be thought of as the long-run average number of events in the time period of interest. To illustrate, if a device manufacturer on average produces 10 defective parts per day, we can model the expected number of defective parts on any given day as $X \sim Poisson(\lambda=10)$. The Poisson is a limiting case of the [[Binomial distribution]] when $n$ gets really large and $p$ gets really small. Some common examples include - Number of winning lottery tickets sold in a given time period. - Number of animals captured in a trap in a given time period. - Number of vehicles crossing a bridge in one day ## Notation $X \sim Poisson(\lambda) $ ## Probability Mass Function $P(X=x) = \frac{\lambda^x}{x!}e^{-\lambda}$ ## Expectation $E(X) = \lambda$ ## Variance $V(X) = \lambda$ ## R notation The Poisson distribution is `pois` in R. ```R # Probability mass function prob <- dpois(x, lambda) # Cumulative distribution function cum_prob <- ppois(q, lambda) # Quantile function quantile_val <- qpois(p, lambda) # Random number generation random_values <- rpois(n, lambda) ```