The poisson distribution models the count of events in a fixed interval when events occur independently and the average rate, \(\lambda\), is constant.

Examples:
Number of emails arriving per minute at a business.
The number of calls received per hour.
Number of defects per unit of product.

The Probability Mass Function is used to give the exact probabilty that a discrete random variable will equal a specific value.
Where for \(X \sim \text{Poisson}(\lambda)\):

\[P(X = k) = \frac{e^{-\lambda}\,\lambda^k}{k!}, \quad k = 0,1,2,\ldots\] Where \(\lambda > 0\) and \(k\) is the observed count

A defining property of the Poisson distribution is that:

\[E[X] = \lambda \qquad \text{Var}(X) = \lambda\] where X is the random variable, \(\lambda\) is the average rate, E[X] is the expected value of the count which equals \(\lambda\), and Var(X) is the variance.
The mean and variance are always equal.

Additionally, another formula, the Moment Generating Function (MGF), can be used to more easily find the moments, such as mean and variance of a distribution, and is:

\[M_X(t) = e^{\,\lambda(e^t - 1)}\]

A business receives \(\lambda = 5\) emails per minute, we can find:

P(exactly 3 emails): \[P(X=3)=\frac{e^{-5}\cdot5^3}{3!}\approx 0.1404\] or
P(at most 4 emails): \[P(X\leq4)=\sum_{k=0}^{4}\frac{e^{-5}\cdot5^k}{k!}\approx 0.4405\]

## [1] 0.1404

## [1] 0.4405

library(plotly)
k_seq   <- 0:35
lam_seq <- seq(0.5, 20, by = 0.5)
z_mat   <- outer(lam_seq, k_seq, function(l, k) dpois(k, l))
p <- plot_ly(x = ~k_seq, y = ~lam_seq, z = ~z_mat,
             type = "surface",
             colorscale = list(c(0,"white"), c(0.5,"purple"), c(1,"black")),
             showscale = FALSE) %>%
  layout(
    margin = list(l = 0, r = 0, t = 10, b = 0),
    scene = list(
      xaxis = list(title = "k"),
      yaxis = list(title = "λ"),
      zaxis = list(title = "P(X = k)")
    )
  )
div(p, style = "width: 800px; height: 450px; margin: 0 auto;")

• Mean = Variance is the main property
• As \(\lambda \to \infty\), Poisson \(\to\) Normal