13  Binomial Distribution

13.1 Definition

The Binomial distribution answers the question: “what is the probability of exactly \(r\) successes after \(n\) independent Bernoulli trials with constant success probability \(p\)?”

\[ \text{P}(X = r) = \begin{cases} \binom{n}{r} p^r q^{n-r}, & r \in \{0,1,\dots,n\} \\ 0, & \text{otherwise} \end{cases} \]

where

\[ \binom{n}{r} = \frac{n!}{(n-r)!r!}, \quad 0 \le r \le n \]

and \(p\) = probability of success, \(q\) = probability of failure, \(p + q = 1\), \(n\) = number of independent draws, and \(X\) = number of successes.

In other words, the Binomial distribution describes the probability of \(X=r\) successes when \(n\) binary experiments are carried out independently with fixed success probability \(p\).

Important contrast: if sampling is done without replacement from a finite population, the exact model is Hypergeometric (see Chapter 16), not Binomial.

13.2 Distribution Function

For integer \(k \in \{0,1,\dots,n\}\):

\[ F(k)=\text{P}(X \le k)=\sum_{r=0}^{k}\binom{n}{r}p^r q^{n-r}. \]

13.3 Mean

\[ \text{E}(X) = n p \]

13.4 Mode

\[ \text{Mo}(X) = \begin{cases} \lfloor (n+1)p \rfloor & \text{if } (n+1)p \notin \mathbb{N} \\ (n+1)p - 1 \text{ and } (n+1)p & \text{if } (n+1)p \in \mathbb{N} \end{cases} \]

When \((n+1)p \in \mathbb{N}\), the distribution is bimodal: two adjacent values are tied for the highest probability.

13.5 Median

\[ \text{Med}(X) \approx np \]

More precisely, a binomial median is in \(\{\lfloor np \rfloor,\lceil np \rceil\}\). In some parameter settings the median is unique; in others both nearby integers satisfy the median condition.

13.6 Variance

\[ \text{V}(X) = n p q \]

Variance is maximal at \(p=0.5\) and approaches 0 as \(p \to 0\) or \(p \to 1\).

13.7 Moment Generating Function

\[ M_X(t) = (q + pe^t)^n \]

13.8 Coefficient of Skewness

\[ g_1 = \frac{q-p}{\sqrt{npq}} \]

13.9 Coefficient of Kurtosis

\[ g_2 = 3 + \frac{1-6pq}{npq} \]

The corresponding excess kurtosis is \(\frac{1-6pq}{npq}\).

13.10 Parameter Estimation

For one observed binomial count \(X\) with known trial size \(n\), the maximum-likelihood estimator is

\[ \hat p = \frac{X}{n}. \]

For a sample \(X_1,\dots,X_m\) of binomial counts with common \((n,p)\):

\[ \hat p = \frac{\bar X}{n}. \]

13.11 R Module

The Binomial Probabilities software can be found on the public website:

The Binomial Probabilities are also available in the menu “Distributions / Binomial Probabilities” of this handbook.

If you prefer to compute Binomial Probabilities on your local computer, the following code can be pasted into the R console:

success_threshold <- 1  # evaluate probabilities at this success count
n_trials <- 10          # number of Bernoulli trials
p_success <- 0.5        # success probability
r <- pbinom(success_threshold, n_trials, p_success)
print('Probabilities of the Binomial Distribution')
print(paste('P(X <= ', success_threshold, ') = ', r, sep=''))
print(paste('P(X > ', success_threshold, ') = ', 1-r, sep=''))
print(paste('P(X = ', success_threshold, ') = ', dbinom(success_threshold, n_trials, p_success), sep=''))
[1] "Probabilities of the Binomial Distribution"
[1] "P(X <= 1) = 0.0107421875"
[1] "P(X > 1) = 0.9892578125"
[1] "P(X = 1) = 0.00976562499999999"

Observe how we define three parameters (success_threshold, n_trials, and p_success) which represent the user input fields of the R module. In fact, all the R scripts displayed in this handbook correspond (to a certain degree) to the R modules that are available on the web.

The main functions that are used are pbinom (distribution function / CDF) and dbinom (probability mass function / PMF). In R, the d prefix is used for both continuous densities and discrete mass functions.

13.12 Example

Let us reconsider the hospital-birth simulation from Section 11.2.1, where sampling variability is compared across different hospital sizes. The number of trials \(n\) is the average number of births in a hospital. The number of “successes” \(r\) to be evaluated is 60% of \(n\). The probability of a “success” is \(p = 0.5\).

For the large hospital \(n = 45\) and \(0.60 n = 0.60 * 45 = 27\). Hence, P\((X \leq 27) \simeq 0.9324\) and P\((X > 27) \simeq 1 - 0.9324 \simeq 0.0676\). You can verify this result with the R console

success_threshold <- 0.6 * 45  # number of successes to be evaluated
n_trials <- 45                 # average number of births in the large hospital
p_success <- 0.5               # probability of a success
r <- pbinom(success_threshold, n_trials, p_success)
print('Probabilities of the Binomial Distribution')
print(paste('P(X <=', success_threshold, ') = ', r, sep=''))
print(paste('P(X >', success_threshold, ') = ', 1-r, sep=''))
print(paste('P(X =', success_threshold, ') = ', dbinom(success_threshold, n_trials, p_success), sep=''))
[1] "Probabilities of the Binomial Distribution"
[1] "P(X <=27) = 0.932421774577165"
[1] "P(X >27) = 0.0675782254228353"
[1] "P(X =27) = 0.0487683705313201"

or with the online R application

Interactive Shiny app (click to load).
Open in new tab

13.13 Additional Academic Example: Vaccine Response Count

In a pilot vaccine study, \(n=30\) participants are independently assessed for seroconversion, with historical response probability \(p=0.6\). What is the probability that at least 22 participants respond?

n_participants <- 30
p_response <- 0.6

cat("P(X >= 22) =",
    1 - pbinom(21, size = n_participants, prob = p_response), "\n")
P(X >= 22) = 0.09401122