Here, we discuss binomial distribution functions in R, plots, parameter setting, random sampling, mass function, cumulative distribution and quantiles.

The binomial distribution for the number of successes in a sequence of independent and identically distributed \(n\) Bernoulli trials, with parameters \(\tt{size}=n\), and \(\tt{probability\:of\:success}=p\) has probability mass function (pmf) formula as:

\[P(X=x)=\binom{n}{x} p^x q^{n-x} = \frac{n!}{x!(n-x)!} p^x q^{n-x},\] for \(x \in \{0, 1, \ldots, n\}\) number of successes,

where \(n \in \{0, 1, 2, \ldots\}\) is the number of trials,

\(p \in [0,1]\) is the probability of success in each trial,

the \(\tt{binomial\; coefficient}\) \(\binom{a}{b}\) equals \(\frac{a!}{b!(a-b)!}\),

and \(a!\), the \(\tt{factorial\; operator}\), equals \(a\times(a-1)\times(a-2)\times\cdots\times 1\).

The mean is \(np\), and the variance is \(npq\).

See also probability distributions and plots and charts.

1 Table of Binomial Distribution Functions in R

The table below shows the functions for binomial distributions in R.

Table of Binomial Distribution Functions in R
Function Usage
rbinom(n, size, prob) Simulate a random sample with \(n\) observations
dbinom(x, size, prob) Calculate the probability mass at the point \(x\)
pbinom(q, size, prob) Calculate the cumulative distribution at the point \(q\)
qbinom(p, size, prob) Calculate the quantile value associated with \(p\)

2 Plot of Binomial Distributions in R

Single distribution:

Below is a plot of the binomial distribution function with \(\tt{size}=20\), and \(\tt{probability\:of\:success}=0.6\).

x = 0:20; y = dbinom(x, 20, 0.6)
plot(x, y, type = "h",
     xlim = c(0, 20), ylim = c(0, max(y)),
     main = "Probability Mass Function of
Binomial Distribution (20, 0.6)",
     xlab = "x", ylab = "Mass",
     col = "blue")
# Add legend
legend("topleft", "size = 20, prob = 0.6",
       fill = "blue",
       bty = "n")
Probability Mass Function (PMF) of a Binomial Distribution in R

Probability Mass Function (PMF) of a Binomial Distribution in R

Multiple distributions:

Below is a plot of multiple binomial distribution functions in one graph.

x1 = 0:30; y1 = dbinom(x1, 30, 0.6)
x2 = 0:30; y2 = dbinom(x2, 30, 0.3)
x3 = 0:20; y3 = dbinom(x3, 20, 0.6)
plot(x1, y1,
     xlim = c(0, 30), ylim = range(c(y1, y2, y3)),
     main = "Probability Mass Functions of Binomial Distributions",
     xlab = "x", ylab = "Mass",
     col = "blue")
points(x2, y2, col = "red")
points(x3, y3, col = "green")
# Add legend
legend("topright", c("size = 30, prob = 0.6",
                    "size = 30, prob = 0.3",
                    "size = 20, prob = 0.6"),
       fill = c("blue", "red", "green"),
       bty = "n")
# Add lines
for(i in 1:30){
  segments(x1[i], dbinom(x1[i], 30, 0.6),
           x1[i+1], dbinom(x1[i+1], 30, 0.6),
           col = "blue")}
for(i in 1:30){
  segments(x2[i], dbinom(x2[i], 30, 0.3),
           x2[i+1], dbinom(x2[i+1], 30, 0.3),
           col = "red")}
for(i in 1:20){
  segments(x3[i], dbinom(x3[i], 20, 0.6),
           x3[i+1], dbinom(x3[i+1], 20, 0.6),
           col = "green")}
Probability Mass Functions (PMFs) of Binomial Distributions in R

Probability Mass Functions (PMFs) of Binomial Distributions in R

3 Examples for Setting Parameters for Binomial Distributions in R

To set the parameters for the binomial distribution function, with \(\tt{size}=10\) and \(\tt{probability\:of\:success}=0.5\).

For example, for dbinom(), the following are the same:

# The order of 10 and 0.5 matters here as the parameter names are not used.
# The first number 10 is size, and 0.5 is prob.
dbinom(2, 10, 0.5)
[1] 0.04394531
# Or:
dbinom(2, size = 10, prob = 0.5)
[1] 0.04394531
# Or:
dbinom(2, prob = 0.5, size = 10)
[1] 0.04394531

4 rbinom(): Random Sampling from Binomial Distributions in R

Sample 800 observations from the binomial distribution with \(\tt{size}=25\) and \(\tt{probability\:of\:success}=0.4\):

rbinom(800, 25, 0.4)
set.seed(12) # Line allows replication (use any number).
sample = rbinom(800, 25, 0.4)
hist(sample,
     main = "Histogram of 800 Observations from
Binomial Distribution with Size = 25 and Prob = 0.4",
     xlab = "x",
     col = "lightblue", border = "white")
Histogram of Binomial Distribution (25, 0.4) Random Sample in R

Histogram of Binomial Distribution (25, 0.4) Random Sample in R

5 dbinom(): Probability Mass Function for Binomial Distributions in R

Calculate the mass at \(x = 14\), in the binomial distribution with \(\tt{size}=18\) and \(\tt{probability\:of\:success}=0.65\):

dbinom(14, 18, 0.65)
[1] 0.1103521
x = 0:18; y = dbinom(x, 18, 0.65)
plot(x, y,
     xlim = c(0, 18), ylim = c(0, max(y)),
     main = "Probability Mass Function of Binomial Distribution
with Size = 18 and Prob = 0.65",
     xlab = "x", ylab = "Mass",
     col = "blue")
# Add lines
segments(14, -1, 14, 0.1103521)
segments(-1, 0.1103521, 14, 0.1103521)
Probability Mass Function (PMF) of Binomial Distribution (18, 0.65) in R

Probability Mass Function (PMF) of Binomial Distribution (18, 0.65) in R

6 pbinom(): Cumulative Distribution Function for Binomial Distributions in R

Calculate the cumulative distribution at \(x = 7\), in the binomial distribution with \(\tt{size}=14\) and \(\tt{probability\:of\:success}=0.4\). That is, \(P(X \le 7)\):

pbinom(7, 14, 0.4)
[1] 0.8498599
x = 0:14; y = pbinom(x, 14, 0.4)
plot(x, y,
     xlim = c(0, 14), ylim = c(0, 1),
     main = "Cumulative Distribution Function of Binomial Distribution
with Size = 14 and Prob = 0.4",
     xlab = "x", ylab = "Cumulative Distribution",
     col = "blue")
# Add lines
for(i in 1:14){
  segments(x[i], pbinom(x[i], 14, 0.4),
           x[i] + 1, pbinom(x[i], 14, 0.4),
           col = "blue")
}
segments(7, -1, 7, 0.8498599)
segments(-1, 0.8498599, 7, 0.8498599)
Cumulative Distribution Function (CDF) of Binomial Distribution (14, 0.4) in R

Cumulative Distribution Function (CDF) of Binomial Distribution (14, 0.4) in R

For upper tail, at \(x = 7\), that is, \(P(X > 7) = 1 - P(X \le 7)\), set the "lower.tail" argument:

pbinom(7, 14, 0.4, lower.tail = FALSE)
[1] 0.1501401

7 qbinom(): Derive Quantile for Binomial Distributions in R

Derive the quantile for \(p = 0.9\), in the binomial distribution with \(\tt{size}=12\) and \(\tt{probability\:of\:success}=0.5\). That is, the \(\tt{smallest}\) \(x\) such that, \(P(X\le x) \ge 0.9\):

qbinom(0.9, 12, 0.5)
[1] 8
x = 0:12; y = pbinom(x, 12, 0.5)
plot(x, y,
     xlim = c(0, 12), ylim = c(0, 1),
     main = "Cumulative Distribution Function of Binomial Distribution
with Size = 12 and Prob = 0.5",
     xlab = "x", ylab = "Cumulative Distribution",
     col = "blue")
# Add lines
for(i in 1:12){
  segments(x[i], pbinom(x[i], 12, 0.5),
           x[i] + 1, pbinom(x[i], 12, 0.5),
           col = "blue")
}
segments(8, -1, 8, 0.9)
segments(-1, 0.9, 8, 0.9)
Cumulative Distribution Function (CDF) of Binomial Distribution (12, 0.5) in R

Cumulative Distribution Function (CDF) of Binomial Distribution (12, 0.5) in R

For upper tail, for \(p = 0.10\), that is, the \(\tt{smallest}\) \(x\) such that, \(P(X > x) < 0.1\):

Note: \(P(X > x) = 1-P(X \le x)\).

qbinom(0.1, 12, 0.5, lower.tail = FALSE)
[1] 8

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