Binomial distribution Measures

Calculates probability mass function (PMF), mean, variance, mode, standard deviation, kurtosis and skewness of a binomial distribution.

Notations

• X : a random variable following a binomial distribution
• n : number of trials
• P(X = k) : probability of getting exactly k successes in n trials
• p : probability of success for each trial (number between 0 and 1)
• q : probability of failure for each trial

q = 1-p
• ([n], [k]) : binomial coefficient n choose k
([n], [k]) = (n!)/(k! * (n-k)!)

Probability Mass Function (PMF)

P(X = k) = ([n], [k]) * p^k*(1-p)^k

Mean (or Expected value)

E(X) = n * p

Standard deviation

sigma(X) = sqrt(n*p*q)

Variance

V(X) = n*p*q

Skewness

S(X) = (q-p)/sqrt(npq)

Kurtosis

K(X) = (1-6*p*q)/sqrt(npq)

Mode

We denote A = (n+1)*p
|__A__| : integer part of A.

• This is the general case, if A = 0 or A is noninteger then,

\text{mode}(X) = |__A__|

• If A is an integer between 1 and n (inclusive),

\text{mode}(X) = A \text{ and } A-1

• If A = n + 1,

\text{mode}(X) = n