# 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`

## See also

Binomial distribution Probabilities

Binomial distribution Histogram

Statistics Calculators