# Uniform Distribution probabilities using R

## Uniform Distribution probabilities using R

In this tutorial, you will learn about how to use dunif(), punif(), qunif() and runif() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Uniform distribution.

Before we discuss R functions for Uniform distribution, let us see what is Uniform distribution.

## Continuous uniform distribution

Uniform distribution is used to real world problems that have uniform probabilistic behavior in a given interval.

A continuous random variable $X$ is said to have a continuous uniform distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by

 \begin{align*} f(x;a,b)&= \begin{cases} \frac{1}{b - a}, & a\leq x\leq b; \\ 0, & Otherwise. \end{cases} \end{align*}

where $a$ and $b$ are the parameters of Uniform distribution.

## Uniform probabilities using dunif() function in R

For continuous probability distribution, density is the value of the probability density function at $x$ (i.e., $f(x)$).

The syntax to compute the probability density function for Uniform distribution using R is

dunif(x,min=0,max=1)

where

• x : the value(s) of the variable and,
• min : lower limit of the distribution (default value is 0 in R),
• max : upper limit of the distribution (default value is 1 in R).

The min and max value must be finite.

Note: If you do not specify the min and max value, R assumes the default value min=0 and max=1 (which is a standard uniform distribution).

The dunif() function gives the value of standard uniform density function for given value(s) of x.

## Numerical Problem for Uniform Distribution

To understand the four functions dunif(), punif(), qunif() and runif(), let us take the following numerical problem.

### Uniform Distribution Example

The daily amount of coffee, in liters, dispensed by a machine located in an airport lobby is a random variable $X$ having a continuous uniform distribution with $a=7$ and $b=10$. Find the probability that on a given day the amount, of coffee dispensed by this machine will be

(a) Find the value of the density function at $x=7.6$.
(b) Plot the graph of Uniform probability distribution.
(c) Find the probability that on a given day the amount of coffee dispensed by the machine will be at most 8.8 liters.
(d) Find the probability that on a given day the amount of coffee dispensed by the machine will be at least 8.5 liters.
(e) Find the probability that on a given day the amount of coffee dispensed by the machine will be more than 7.4 liters but less than 9.5 liters.
(f) Plot the graph of cumulative Uniform probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.60$?
(h) Simulate 1000 Uniform distributed random variables with $a=7$ and $b=10$.

Let $X$ denote the daily amount of coffee, in liters, dispensed by a machine. Given that $X\sim U(a=7,b=10)$.

### Example 1: How to use  dunif() function in R?

To find the value of the density function at $x=2.5$ we need to use  dunif() function.

First let us define the given parameters as

# lower limit of the distribution
a <- 7
# upper limit of the distribution
b <- 10

The probability density function of $X$ is

 \begin{aligned} f(x)&= \frac{1}{10 - 7},\\ &\quad\text{for }7 \leq x \leq 10. \end{aligned}

For part (a), we need to find the density function at $x=7.6$. That is $f(7.6)$.

(a) The value of the density function at $x=7.6$ is

 \begin{aligned} f(7.6)&= \frac{1}{10-7}\\ &=0.3333333\\ &= 0.3333333 \end{aligned}

The above probability can be calculated using dunif(7.6,min=7,max=10) function in R.

# Compute Uniform probability
result1 <- dunif(7.6,min=a,max=b)
result1
[1] 0.3333333

### Example 2 Visualize Uniform probability distribution

Using dunif() function we can compute Uniform distribution probabilities for given x, min and max. To plot the probability density function of Uniform distribution, we need to create a sequence of x values and compute the corresponding probabilities.

# create a sequence of x values
x <- seq(7,10, by=0.02)
## Compute the Uniform pdf for each x
px<- dunif(x,min=a,max=b)

(b) Visualizing Uniform Distribution with dunif() function and plot() function in R:

The probability density function of Uniform distribution with given 7 and 10 can be visualized using plot() function as follows:

## Plot the Uniform probability dist
plot(x,px,type="l",xlim=c(6,11),ylim=c(0,0.5),
lwd=3, col="red",ylab="f(x)")
title("PDF of Uniform (a=7,b=10)")
polygon(c(7,x,10),c(0,px,0),col="lightgray",border=NA)
lines(x,px,type = "l",lwd=2,col="red")

## Uniform cumulative probability using punif() function in R

The syntax to compute the cumulative probability distribution function (CDF) for Uniform distribution using R is

punif(q, min=0,max=1)

where

• q : the value(s) of the variable,
• min : lower limit of the distribution (default value is 0 in R),
• max : upper limit of the distribution (default value is 1 in R).

Using this function one can calculate the cumulative distribution function of standard uniform distribution for given value(s) of q (value of the variable x).

### Example 3: How to use punif() function in R?

In the above example, for part (c), we need to find the probability $P(X\leq 8.8)$.

(c) The probability that on a given day the amount of coffee dispensed by the machine will be at most 8.8 liters is

 \begin{aligned} P(X\leq 8.8) &=\int_7^{8.8} f(x)\; dx\\ &=\frac{1}{3}\int_7^{8.8}\; dx\\ &=\frac{1}{3}[x]_7^{8.8}\\ &=\frac{1}{3}[8.8 - 7]\\ &=0.6. \end{aligned}

The same probability can be computed using R function punif() as follows:

## Compute cumulative Uniform probability
result2 <- punif(8.8,min=a,max=b)
result2
[1] 0.6

### Example 4: How to use punif() function in R?

In the above example, for part (d), we need to find the probability $P(X \geq 8.5)$.

(d) The probability that on a given day the amount of coffee dispensed by the machine will be at least 8.5 liters is

 \begin{aligned} P(X\geq 8.5) &=\int_{8.5}^{10} f(x)\; dx\\ &=\frac{1}{3}\int_{8.5}^{10} \; dx\\ &=\frac{1}{3}[10-8.5] \\ &=0.5. \end{aligned}

To calculate the probability that a random variable $X$ is greater than a given number, one can use the option lower.tail=FALSE in punif() function.

Above probability can be calculated easily using punif() function with argument lower.tail=FALSE as

$P(X \geq 8.5) =\int_{8.5}^{10} f(x)\; dx$= punif(8.5,min=a,max=b,lower.tail=FALSE)

# compute cumulative Uniform probabilities
# with lower.tail False
punif(8.5,min=a,max=b,lower.tail=FALSE)
[1] 0.5

or by using complementary event as

$P(X \geq 8.5) = 1- P(X\leq 8.5)$= 1- punif(8.5,min=a,max=b)

# Using complementary event
1-punif(8.5,min=a,max=b)
[1] 0.5

### Example 5: How to use punif() function in R?

One can also use punif() function to calculate the probability that the random variable $X$ is between two values.

(e) The probability that on a given day the amount of coffee dispensed by the machine will be more than 7.4 liters but less than 9.5 liters can be written as $P(7.4 < X < 9.5)$.

 \begin{aligned} P(7.4 < X < 9.5) &= P(X< 9.5) -P(X < 7.4)\\ &= 0.8333333 - 0.1333333\\ &= 0.7 \end{aligned}

The above probability can be calculated using punif() function as follows:

res1 <- punif(9.5,min=a,max=b)
res2 <- punif(7.4,min=a,max=b)
result3 <- res1 - res2
result3
[1] 0.7

### Example 6: Visualize the cumulative Uniform probability distribution

Using punif() function we can compute Uniform cumulative probabilities (CDF) for given x and rate. To plot the CDF of Uniform distribution, we need to create a sequence of x values and compute the corresponding cumulative probabilities.

# create a sequence of x values
x <- seq(7,10, by=0.02)
## Compute the Uniform pdf for each x
Fx <- punif(x,min=a,max=b)

(f) Visualizing Uniform Distribution with punif() function and plot() function in R:

The cumulative probability distribution of Uniform distribution with given x, min and max can be visualized using plot() function as follows:

## Plot the Uniform  probability dist
plot(x,Fx,type="l",xlim=c(6,11),ylim=c(0,1),
lwd=3, col="red",ylab="F(x)")
title("CDF of U(7,10)")
polygon(c(7,x,10),c(0,Fx,0),col="lightgray",border=NA)
lines(x,Fx,type = "l",lwd=2,col="red")

## Uniform Distribution Quantiles using qunif() in R

The syntax to compute the quantiles of Uniform distribution using R is

qunif(p,min=0,max=1)

where

• p : the value(s) of the probabilities,
• min : lower limit of the distribution (default value is 0 in R),
• max : upper limit of the distribution (default value is 1 in R).

The function qunif(p,min=0,max=1) gives $100*p^{th}$ quantile of standard uniform distribution for given value of p.

The $p^{th}$ quantile is the smallest value of Uniform random variable $X$ such that $P(X\leq x) \geq p$.

It is the inverse of punif() function. That is, inverse cumulative probability distribution function for Uniform distribution.

### Example 7: How to use qunif() function in R?

In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.60$. That is we need to find the $60^{th}$ quantile of given Uniform distribution.

a <- 7
b<- 10
prob <- 0.60
# compute the quantile for Uniform  dist
qunif(0.60,min=a,max=b)
[1] 8.8

The $60^{th}$ percentile of given Uniform distribution is 8.8.

### Visualize the quantiles of Uniform Distribution

The quantiles of Uniform distribution with given p, min and max can be visualized using plot() function as follows:

p <- seq(0,1,by=0.02)
qx <- qunif(p,min=a,max=b)
# Plot the Quantiles of Uniform  dist
plot(p,qx,type="l",lwd=2,col="darkred",
ylab="quantiles",
main="Quantiles of Uniform(7,10)")

## Simulating Uniform random variable using runif() function in R

The general R function to generate random numbers from Uniform distribution is

runif(n,min=0,max=1)

where,

• n : the sample observations,
• min : lower limit of the distribution (default value is 0 in R),
• max : upper limit of the distribution (default value is 1 in R).

The function runif(n,min=0,max=1) generates n random numbers from standard uniform distribution.

### Example 8: How to use runif() function in R?

In part (h), we need to generate 1000 random numbers from Uniform distribution with given parameters $min = 7,max=10$.

(h) We can use runif(1000,min=a,max=b) function to generate random numbers from Uniform $U(7,10)$ distribution.

## initialize sample size to generate
n <- 1000
# Simulate 1000 values From Uniform  dist
x_sim <- runif(n,min=a,max=b)

The below graphs shows the density of the simulated random variables from Uniform $U(7,10)$ Distribution.

## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Uniform(7,10) dist")

If you use same function again, R will generate another set of random numbers from $U(7,10)$.

# Simulate 1000 values From Uniform  dist
x_sim_2 <- runif(n,min=a,max=b)
## Plot the simulated data
plot(density(x_sim_2),xlab="Simulated x",ylab="density",
lwd=5,col="blue",
main="Simulated data from U(7,10) dist")

For the simulation purpose to reproduce same set of random numbers, one can use set.seed() function.

# set seed for reproducibility
set.seed(1457)
# Simulate 1000 values From Uniform  dist
x_sim_3 <- runif(n,min=a,max=b)
## Plot the simulated data
plot(density(x_sim_3),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from U(7,10) dist")
set.seed(1457)
# Simulate 1000 values From Uniform  dist
x_sim_4 <- runif(n,min=a,max=b)
## Plot the simulated data
plot(density(x_sim_4),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from U(7,10) dist")

Since we have used set.seed(1457) function, R will generate the same set of Uniform distributed random numbers.

The histogram of randomly simulated values from $U(7,10)$ distribution as as follows:

hist(x_sim_4,breaks = 30,
main="Simulated Values from U(7,10)")

To learn more about other discrete and continuous probability distributions using R, go through the following tutorials:

Discrete Distributions Using R

Continuous Distributions Using R

## Endnote

In this tutorial, you learned about how to compute the probabilities, cumulative probabilities and quantiles of Uniform distribution in R programming. You also learned about how to simulate a Uniform distribution using R programming.