Documentation

Mathlib.Probability.ProbabilityMassFunction.Binomial

The binomial distribution #

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

Main results #

noncomputable def PMF.binomial (p : ENNReal) (h : p 1) (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”.

Equations
Instances For
    theorem PMF.binomial_apply (p : ENNReal) (h : p 1) (n : ) (i : Fin (n + 1)) :
    (PMF.binomial p h n) i = p ^ i * (1 - p) ^ ((Fin.last n) - i) * (n.choose i)
    @[simp]
    theorem PMF.binomial_apply_zero (p : ENNReal) (h : p 1) (n : ) :
    (PMF.binomial p h n) 0 = (1 - p) ^ n
    @[simp]
    theorem PMF.binomial_apply_last (p : ENNReal) (h : p 1) (n : ) :
    (PMF.binomial p h n) (Fin.last n) = p ^ n
    theorem PMF.binomial_apply_self (p : ENNReal) (h : p 1) (n : ) :
    (PMF.binomial p h n) n = p ^ n
    theorem PMF.binomial_one_eq_bernoulli (p : ENNReal) (h : p 1) :
    PMF.binomial p h 1 = PMF.map (fun (x : Bool) => bif x then 1 else 0) (PMF.bernoulli p h)

    The binomial distribution on one coin is the bernoully distribution.