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The binomial distribution arises when n independent Bernoulli trials are conducted, each with the same probability of success p. The number of successes is a binomial(n, p) random variable.

The function ⌊x⌋ in the cdf denotes the "floor" or greatest integer function.

Parameter Range Description
n n = 1, 2, ... Number of trials
p 0 ≤ p ≤ 1 Probability of success

Probability Mass Function

P ( X = x ; n , p ) = ( n x ) p x ( 1 - p ) n - x
F ( x ; n , p ) = ∑ i = 0 ⌊ x ⌋ ( n i ) p i ( 1 - p ) n - i

Support

x = 0 , 1 , … , n

Mean

Variance

Example n p
A fair six-sided die is thrown 3 times. Let X be the number of sixes thrown. 3 0.1667
The probability that a pregnancy in the US will result in a girl is 48.8%. A family has 7 children. Let X be the number of girls. 7 0.4880
For a particular student, each of the 58 questions on the math SAT has an 80% chance of being answered correctly. Let X be the number of correct answers. 58 0.8000

X ∼ Binomial(n, p)

n = p =
Chart of the binomial distribution Chart area for displaying the binomial pmf, cdf, visualizations, and simulations

E(X) = , Var(X) =

Note that, since a Bernoulli random variable equals 1 for a success and 0 for a failure, then a binomial random variable is the sum of n independent Bernoulli random variables. The mean and variance are therefore n times the mean and variance of a Bernoulli random variable (which are p and p(1 − p)).

Note that, since a Bernoulli random variable equals 1 for a success and 0 for a failure, then a binomial random variable is the sum of n independent Bernoulli random variables. The mean and variance are therefore n times the mean and variance of a Bernoulli random variable (which are p and p(1 − p)).

The illustration above shows a set of n independent Bernoulli(p) trials. The random variable X is the total number of successes, which has a binomial(n, p) distribution.

The simulation above shows a set of n independent Bernoulli(p) trials. Successful trials are shown in green and failures in grey. The random variable X is the number of successes, which has a binomial(n, p) distribution. The histogram accumulates the results of each simulation.

Y = Binomial(1, p) ∼ Bernoulli(p) ∑ Xᵢ ∼ Binomial(Σnᵢ, p) limn→∞ X ∼ Normal(np, np(1 − p)) limn→∞, np=λ X ∼ Poisson(np)

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

The binomial(n, p) distribution is shown in red superimposed over the Bernoulli . It can be seen that these are equal when n = 1. The reason is that, when n = 1, the binomial distribution counts the number of successes in a single Bernoulli(p) trial. This is either 1, with probability p, or 0, with probability 1 − p.

This can be thought of as counting the number of successes in n₁ + ··· + nₘ independent Bernoulli(p) trials, which is the definition of a binomial(n₁ + ··· + nₘ, p) random variable.

The graph above shows the case when n₁ = n₂ = ··· = nₘ.

Note that a binomial(n, p) random variable is the sum of n independent Bernoulli(p) random variables with mean p and variance p(1 − p). By the Central Limit Theorem, the sum of n i.i.d random variables with finite variance converges to a normal random variable as n → ∞. So, the sum converges to a normal(np, np(1 − p)) random variable.

The binomial distribution is shown in red superimposed over the normal limit.

For small values of p, the Poisson approximation shown above is quite good even for small values of n. As p → 0, the variance np(1 − p) of the binomial converges to the mean np. Since the mean and variance of the Poisson are identical, the binomial looks more like the Poisson as p gets smaller.

The binomial distribution is shown in red superimposed over the Poisson limit.