Binomial random variable

• Definition: Let $X$ be a binomial random variable, then:

  • $p_X(k) = {n \choose k}p^k (1-p)^{n-k}$, where $p$ is a fixed value in $(0, 1)$ called the binomial parameter, $n$ is the total number of states, and $k$ belongs in $\left\{ 0, 1, \dots, n \right\}$

  • Notation: $X \sim Binomial(n, p)$

• Theorem: Let $X \sim Binomial(n, p)$, then:
  • $\mathbb{E}[X] = np$
  • $\mathbb{E}[X^2] = np(np + (1-p))$
  • $Var[X] = np\cdot (1-p)$

n <- 10
states <- 0:n
p <- 0.3

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