Poisson random variable

• Definition: Let \(X\) be a Poisson random variable, then:
  • \(p_X(k) = \dfrac{\lambda^k}{k!}e^{-\lambda}\), where \(\lambda\) is a fixed positive value called the Poisson rate, and \(k\) is a non-negative integer.
  • Notation: \(X \sim Poisson(\lambda)\)
• Theorem: Let \(X \sim Poisson(\lambda)\), then:
  • \(\mathbb{E}[X] = \lambda\)
  • \(\mathbb{E}[X^2] = \lambda + \lambda^2\)
  • \(Var[X] = \lambda\)

n <- 20
states <- 0:n
lambda <- 1

plot(
  states, dpois(states, lambda), type = 'h',
  ylim = c(0, 1), main = "Poisson(lambda)",
  xlab = "States", ylab = "Probabilities"
)