The binomial distribution

This file defines the probability mass function of the binomial distribution.

Main results

• binomial_one_eq_bernoulli: For n = 1, it is equal to PMF.bernoulli.

def PMF.binomial (p : NNReal) (h : p ≤ 1) (n : ℕ) : PMF (Fin (n + 1))

The binomial PMF: the probability of observing exactly i “heads” in a sequence of n independent coin tosses, each having probability p of coming up “heads”.

theorem PMF.binomial_apply (p : NNReal) (h : p ≤ 1) (n : ℕ) (i : Fin (n + 1)) : (binomial p h n) i = ↑p ^ ↑i * (1 - ↑p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)

@[simp]

theorem PMF.binomial_apply_zero (p : NNReal) (h : p ≤ 1) (n : ℕ) : (binomial p h n) 0 = (1 - ↑p) ^ n

@[simp]

theorem PMF.binomial_apply_last (p : NNReal) (h : p ≤ 1) (n : ℕ) : (binomial p h n) (Fin.last n) = ↑p ^ n

theorem PMF.binomial_apply_self (p : NNReal) (h : p ≤ 1) (n : ℕ) : (binomial p h n) (Fin.last n) = ↑p ^ n

theorem PMF.binomial_one_eq_bernoulli (p : NNReal) (h : p ≤ 1) : binomial p h 1 = map (fun (x : Bool) => bif x then 1 else 0) (bernoulli p h)

The binomial distribution on one coin is the Bernoulli distribution.

theorem PMF.binomial_apply_of_le {k b : ℕ} (hb : k ≤ b) {x : NNReal} (h : x ≤ 1) : ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) = (binomial x h b) (Fin.ofNat (b + 1) k)