Geometric random variable

• Definition: Let $X$ be a geometric random variable, then:
  • $p_X(k) = (1-p)^{k-1}p$, where $p$ is a fixed value in $(0, 1)$ called the geometric parameter and $k$ is a positive integer.
  • Notation: $X \sim Geometric(p)$
• Theorem: Let $X \sim Geometric(p)$, then:
  • $\mathbb{E}[X] = 1/p$
  • $\mathbb{E}[X^2] = 2/p^2 - 1/p$
  • $Var[X] = \dfrac{1-p}{p^2}$

n <- 20
states <- 1:n
p <- 0.4

plot(
  states, dgeom(states, p), type = 'h',
  ylim = c(0, 1), main = "Geom(p)",
  xlab = "States", ylab = "Probabilities"
)