Here, we discuss continuous uniform distribution functions in R, plots, parameter setting, random sampling, density, cumulative distribution and quantiles.

The continuous uniform distribution with parameters \(\tt{min}=a\), and \(\tt{max}=b\) has probability density function (pdf) formula as:

\[f(x)=\begin{cases} \frac{1}{b - a} & \mathrm{for}\ a \le x \le b, \\[8pt] 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b \end{cases},\] where \(-\infty < a < b < \infty\).

The mean is \(\frac{1}{2}(a+b)\), and the variance is \(\frac{1}{12}(b-a)^2\).

See also probability distributions and plots and charts.

1 Table of Continuous Uniform Distribution Functions in R

The table below shows the functions for continuous uniform distributions in R.

Table of Continuous Uniform Distribution Functions in R
Function Usage
runif(n, min=0, max=1) Simulate a random sample with \(n\) observations
dunif(x, min=0, max=1) Calculate the probability density at the point \(x\)
punif(q, min=0, max=1) Calculate the cumulative distribution at the point \(q\)
qunif(p, min=0, max=1) Calculate the quantile value associated with \(p\)

2 Plot of Continuous Uniform Distributions in R

Below is a plot of the continuous uniform distribution function with \(\min=0\) and \(\max=1\).

x = seq(0, 1); y = dunif(x, 0, 1)
plot(x, y, type = "l",
     xlim = c(-1, 2), ylim = c(0, 1.2),
     main = "Probability Density Function of
Continuous Uniform Distribution (0, 1)",
     xlab = "x", ylab = "Density",
     lwd = 4, col = "blue")
# Add lines
segments(-1, 0, 0, 0, col = "blue")
segments(0, 0, 0, 1, lty = "dashed", col = "blue")
segments(1, 1, 1, 0, lty = "dashed", col = "blue")
segments(1, 0, 2, 0, col = "blue")
Probability Density Function (PDF) of a Continuous Uniform Distribution in R

Probability Density Function (PDF) of a Continuous Uniform Distribution in R

3 Examples for Setting Parameters for Continuous Uniform Distributions in R

In the continuous uniform distribution functions, parameters are pre-specified as \(\min=0\) and \(\max=1\), hence they do not need to be specified, unless they are to be set to different values.

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

# The order of 0 and 1 matters here as the parameter names are not used.
# The first number 0 is min, and 1 is max.
punif(0.4); punif(0.4, 0); punif(0.4, 0, 1)
[1] 0.4
[1] 0.4
[1] 0.4
# Or:
punif(0.4, min = 0, max = 1)
[1] 0.4
# Or:
punif(0.4, max = 1, min = 0)
[1] 0.4

4 runif(): Random Sampling from Continuous Uniform Distributions in R

Sample 500 observations from the continuous uniform distribution with \(\min = 2\) and \(\max = 4\):

runif(500, 2, 4)
set.seed(1234) # Line allows replication (use any number).
sample = runif(500, 2, 4)
hist(sample,
     main = "Histogram of 500 Observations from
Continuous Uniform Distribution with Min = 2 and Max = 4",
     xlab = "x",
     col = "deepskyblue", border = "white")
Histogram of Continuous Uniform Distribution (2, 4) Random Sample in R

Histogram of Continuous Uniform Distribution (2, 4) Random Sample in R

5 dunif(): Probability Density Function for Continuous Uniform Distributions in R

Calculate the density at \(x = 1.5\), in the continuous uniform distribution with \(\min = 1\) and \(\max = 3\):

dunif(1.5, 1, 3)
[1] 0.5
x = seq(1, 3); y = dunif(x, 1, 3)
plot(x, y, type = "l",
     xlim = c(0, 4), ylim = c(0, 0.6),
     main = "Probability Density Function of
Continuous Uniform Distribution with Min = 1 and Max = 3",
     xlab = "x", ylab = "Density",
     lwd = 4, col = "blue")
# Add lines
segments(0, 0, 1, 0, col = "blue")
segments(1, 0, 1, 0.5, lty = "dashed", col = "blue")
segments(3, 0.5, 3, 0, lty = "dashed", col = "blue")
segments(3, 0, 4, 0, col = "blue")

segments(1.5, -1, 1.5, 0.5, lty = "dashed")
segments(-1, 0.5, 1.5, 0.5, lty = "dashed")
Probability Density Function (PDF) of Continuous Uniform Distribution (1, 3) in R

Probability Density Function (PDF) of Continuous Uniform Distribution (1, 3) in R

6 punif(): Cumulative Distribution Function for Continuous Uniform Distributions in R

Calculate the cumulative distribution at \(x = 3.5\), in the continuous uniform distribution with \(\min = 2\) and \(\max = 6\). That is, \(P(X \le 3.5)\):

punif(3.5, 2, 6)
[1] 0.375
x = seq(2, 6); y = punif(x, 2, 6)
plot(x, y, type = "l",
     xlim = c(1, 7), ylim = c(0,1),
     main = "Cumulative Distribution Function of
Continuous Uniform Distribution with Min = 2 and Max = 6",
     xlab = "x", ylab = "Cumulative Distribution",
     lwd = 4, col = "blue")
# Add lines
segments(1, 0, 2, 0, col = "blue")
segments(6, 1, 7, 1, col = "blue")

segments(3.5, -1, 3.5, 0.375, lty = "dashed")
segments(-1, 0.375, 3.5, 0.375, lty = "dashed")
Cumulative Distribution Function (CDF) of Continuous Uniform Distribution (2, 6) in R

Cumulative Distribution Function (CDF) of Continuous Uniform Distribution (2, 6) in R 

x = seq(2, 6); y = dunif(x, 2, 6)
plot(x, y, type = "l",
     xlim = c(1, 7), ylim = c(0, 0.3),
     main = "Probability Density Function of
Continuous Uniform Distribution with Min = 2 and Max = 6",
     xlab = "x", ylab = "Density",
     lwd = 4, col = "blue")
# Add lines
segments(1, 0, 2, 0, col = "blue")
segments(2, 0, 2, 0.25, lty = "dashed", col = "blue")
segments(6, 0.25, 6, 0, lty = "dashed", col = "blue")
segments(6, 0, 7, 0, col = "blue")
# Add shaded region and legend
polygon(x = c(2, 2, 3.5, 3.5),
        y = c(0, 0.25, 0.25, 0),
        col = "limegreen")
legend("topright", c("Area = 0.375"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Probability Density Function (PDF) of Continuous Uniform Distribution (2, 6) in R

Shaded Probability Density Function (PDF) of Continuous Uniform Distribution (2, 6) in R

For upper tail, at \(x = 3.5\), that is, \(P(X \ge 3.5) = 1 - P(X \le 3.5)\), set the "lower.tail" argument:

punif(3.5, 2, 6, lower.tail = FALSE)
[1] 0.625
x = seq(2, 6); y = dunif(x, 2, 6)
plot(x, y, type = "l",
     xlim = c(1, 7), ylim = c(0, 0.3),
     main = "Shaded Upper Region: Probability Density Function of
Continuous Uniform Distribution with Min = 2 and Max = 6",
     xlab = "x", ylab = "Density",
     lwd = 4, col = "blue")
# Add lines
segments(1, 0, 2, 0, col = "blue")
segments(2, 0, 2, 0.25, lty = "dashed", col = "blue")
segments(6, 0.25, 6, 0, lty = "dashed", col = "blue")
segments(6, 0, 7, 0, col = "blue")
# Add shaded region and legend
polygon(x = c(3.5, 3.5, 6, 6),
        y = c(0, 0.25, 0.25, 0),
        col = "limegreen")
legend("topright", c("Area = 0.625"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Upper Region: Probability Density Function (PDF) of Continuous Uniform Distribution (2, 6) in R

Shaded Upper Region: Probability Density Function (PDF) of Continuous Uniform Distribution (2, 6) in R

7 qunif(): Derive Quantile for Continuous Uniform Distributions in R

Derive the quantile for \(p = 0.6\), in the continuous uniform distribution with \(\min = 0\) and \(\max = 2\). That is, \(x\) such that, \(P(X \le x)=0.6\):

qunif(0.6, 0, 2)
[1] 1.2
x = seq(0, 2); y = punif(x, 0, 2)
plot(x, y, type = "l",
     xlim = c(-1, 3), ylim = c(0,1),
     main = "Cumulative Distribution Function of
Continuous Uniform Distribution with Min = 0 and Max = 2",
     xlab = "x", ylab = "Cumulative Distribution",
     lwd = 4, col = "blue")
# Add lines
segments(-1, 0, 0, 0, col = "blue")
segments(2, 1, 3, 1, col = "blue")

segments(1.2, -1, 1.2, 0.6, lty = "dashed")
segments(-2, 0.6, 1.2, 0.6, lty = "dashed")
Cumulative Distribution Function (CDF) of Continuous Uniform Distribution (0, 2) in R

Cumulative Distribution Function (CDF) of Continuous Uniform Distribution (0, 2) in R 

x = seq(0, 2); y = dunif(x, 0, 2)
plot(x, y, type = "l",
     xlim = c(-1, 3), ylim = c(0, 0.6),
     main = "Probability Density Function of
Continuous Uniform Distribution with Min = 0 and Max = 2",
     xlab = "x", ylab = "Density",
     lwd = 4, col = "blue")
# Add lines
segments(-1, 0, 0, 0, col = "blue")
segments(0, 0, 0, 0.5, lty = "dashed", col = "blue")
segments(2, 0.5, 2, 0, lty = "dashed", col = "blue")
segments(2, 0, 3, 0, col = "blue")
# Add shaded region and legend
polygon(x = c(0, 0, 1.2, 1.2),
        y = c(0, 0.5, 0.5, 0),
        col = "limegreen")
legend("topright", c("Area = 0.6"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Probability Density Function (PDF) of Continuous Uniform Distribution (0, 2) in R

Shaded Probability Density Function (PDF) of Continuous Uniform Distribution (0, 2) in R

For upper tail, for \(p = 0.4\), that is, \(x\) such that, \(P(X \ge x)=0.4\):

qunif(0.4, 0, 2, lower.tail = FALSE)
[1] 1.2
x = seq(0, 2); y = dunif(x, 0, 2)
plot(x, y, type = "l",
     xlim = c(-1, 3), ylim = c(0, 0.6),
     main = "Shaded Upper Region: Probability Density Function of
Continuous Uniform Distribution with Min = 0 and Max = 2",
     xlab = "x", ylab = "Density",
     lwd = 4, col = "blue")
# Add lines
segments(-1, 0, 0, 0, col = "blue")
segments(0, 0, 0, 0.5, lty = "dashed", col = "blue")
segments(2, 0.5, 2, 0, lty = "dashed", col = "blue")
segments(2, 0, 3, 0, col = "blue")
# Add shaded region and legend
polygon(x = c(1.2, 1.2, 2, 2),
        y = c(0.5, 0, 0, 0.5),
        col = "limegreen")
legend("topright", c("Area = 0.4"),
       fill = c("limegreen"),
       inset = 0.01)
Shaded Upper Region: Probability Density Function (PDF) of Continuous Uniform Distribution (0, 2) in R

Shaded Upper Region: Probability Density Function (PDF) of Continuous Uniform Distribution (0, 2) in R

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