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"
)