Laplace Distribution Probabilities Using R

Laplace Distribution probabilities using R

In this tutorial, you will learn about how to use dlaplace(), plaplace(), qlaplace() and rlaplace() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Laplace distribution.

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

Laplace Distribution

Laplace distribution distribution is a continuous type probability distribution.

Let $X\sim L(\mu,\lambda)$. Then the probability distribution of $X$ is

$$ \begin{align*} f(x;\mu, \lambda)&= \begin{cases} \frac{\lambda}{2}e^{-\lambda|x-\mu|}, & -\infty < x< \infty; \\ & -\infty < \mu < \infty, \lambda >0; \\ 0, & Otherwise. \end{cases} \end{align*} $$

where $\mu$ is the location parameter and $\lambda$ is the scale parameter of Laplace distribution.

The distribution function of Laplace distribution is

$$ \begin{align*} F(x) &= \begin{cases} \frac{1}{2}e^{\lambda(x-\mu)}, & x< \mu; \\ &\\ 1-\frac{1}{2}e^{-\lambda(x-\mu)}, & x\geq \mu; \end{cases} \end{align*} $$

Read more about the theory and results of Laplace distribution here.

Laplace probabilities using dlaplace() function in R

The Laplace distribution is not available in base package. It is available in VGAM (Vector Generalized Linear and Additive Models) package.

First we need to install the package using install.packages(VGAM), if it is not installed. Then to use the functions from VGAM package we need to load it using library(VGAM) command.

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 Laplace distribution using R is

dlaplace(x,location, scale)

where

• x : the value(s) of the variable and,
• location : location parameter of Laplace distribution,
• scale : scale parameter of Laplace distribution.

The dlaplace() function gives the density for given value(s) x, location and scale.

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

Numerical Problem for Laplace Distribution

To understand the four functions dlaplace(), plaplace(), qlaplace() and rlaplace(), let us take the following numerical problem.

Laplace Distribution Example

Let $X$ be a random variable with Laplace distribution with parameter $\mu=5$ and $\lambda=2$. That is $X\sim L(5,2)$ distribution.

(a) Find the value of the density function at $x=3.5$.
(b) Plot the graph of Laplace probability distribution.
(c) Find the probability that, $X$ is less than or equal to 6.
(d) Find the probability that, $X$ is greater than 3.5.
(e) Find the probability that, $X$ is less than 10 but greater than 6.
(f) Plot the graph of cumulative Laplace probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.75$?
(h) Simulate 1000 Laplace distributed random variables with $\mu= 5$ and $\lambda = 2$.

Given that $X\sim L(\mu=5, \lambda=2)$.

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

To find the value of the density function at $x=0.35$ we need to use dlaplace() function.

library(VGAM)
# parameter 1 location
mu <- 5
# parameter 2 scale
lambda <- 2

The probability density function of $X$ is

$$ \begin{aligned} f(x)&= \frac{2}{2}e^{-2|x-5|},\\ &\quad\text{for } -\infty \leq x \leq \infty. \end{aligned} $$

For part (a), we need to find the density function at $x=3.5$. That is $f(3.5)$.

First I will show you how to calculate the value of the density function for given value of $x$. Then I will show you how to compute the same using dlaplace() function in R.

(a) The value of the density function at $x=3.5$ is

$$ \begin{aligned} f(3.5)&= \frac{2}{2}e^{-2|3.5-5|}\\ &= 1\times e^{-3}\\ &= 0.1180916 \end{aligned} $$

The above probability can be calculated using dlaplace(3.5,5,2) function in R.

# Compute Laplace  probability
result1 <- dlaplace(3.5,mu,lambda)
result1

[1] 0.1180916

Example 2 Visualize Laplace probability distribution

Using dlaplace() function we can compute Laplace distribution probabilities for given x, location and scale. To plot the probability density function of Laplace distribution, we need to create a sequence of x values and compute the corresponding probabilities.

# create a sequence of x values
x <- seq(-5,15, by=0.02)
## Compute the Laplace pdf for each x
px<-dlaplace(x,mu,lambda)

(b) Visualizing Laplace Distribution with dlaplace() function and plot() function in R:

The probability density function of Laplace distribution with given 5 and 2 can be visualized using plot() function as follows:

## Plot the Laplace  probability dist
plot(x,px,type="l",xlim=c(-5,15),ylim=c(0,max(px)),
     lwd=3, col="darkred",ylab="f(x)",
     main=expression(paste("PDF of L(",
        mu,"=5, ",lambda,"=2)")))

Laplace cumulative probability using plaplace() function in R

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

plaplace(q,location, scale)

where

• q : the value(s) of the variable,
• location : first parameter of Laplace distribution,
• scale : second parameter of Laplace distribution.

Using this function one can calculate the cumulative distribution function of Laplace distribution for given value(s) of q (value of the variable x), location and scale.

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

In the above example, for part (c), we need to find the probability $P(X\leq 6)$.

(c) The probability that, $X$ is less than or equal to 6 is

$$ \begin{aligned} P(X \leq 6) &=F(6)\\ &=1-\frac{1}{2}e^{\dfrac{-(6-5)}{2}}\\ &\qquad (\because 6 > \mu)\\ &=1-\frac{1}{2}e^{\dfrac{-(6-5)}{2}}\\ &= 1-0.3033\\ &= 0.6967 \end{aligned} $$

## Compute cumulative Laplace  probability
result2 <- plaplace(6,mu,lambda)
result2

[1] 0.6967347

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

In the above example, for part (d), we need to find the probability $P(X \geq 3.5)$.

(d) The probability that $X$ is greater than $3.5$ is

$$ \begin{aligned} P(X > 3.5) &=1-P(X < 3.5)\\ &=1-F(3.5)\\ &=1-\frac{1}{2}e^{\dfrac{-(3.5-5)}{2}}\\ &\qquad (\because 3.5 < \mu)\\ &=1-\frac{1}{2}e^{\dfrac{(3.5-5)}{2}}\\ &= 1-0.2362\\ &= 0.7638 \end{aligned} $$

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

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

$P(X > 3.5) =$ plaplace(3.5,mu,lambda,lower.tail=FALSE)

or by using complementary event as

$P(X > 3.5) = 1- P(X\leq 3.5)$= 1- plaplace(3.5,mu,lambda)

# compute cumulative Laplace  probabilities
# with lower.tail False
plaplace(3.5,mu,lambda,lower.tail=FALSE)

[1] 0.7638167

# Using complementary event
1-plaplace(3.5,mu,lambda)

[1] 0.7638167

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

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

(e) The probability that $X$ is between $6$ and $10$ is

$$ \begin{aligned} P(6 \leq X \leq 10)&=P(X\leq 10)-P(X\leq 6)\\ &=F(10) -F(6)\\ &=\bigg(1-\frac{1}{2}e^{\dfrac{-(10-5)}{2}}\bigg)-\bigg(1-\frac{1}{2}e^{\dfrac{-(6-5)}{2}}\bigg)\\ &=\frac{1}{2}e^{\dfrac{-(6-5)}{2}}-\frac{1}{2}e^{\dfrac{-(10-5)}{2}}\\ &= 0.3033-0.041\\ &=0.2623 \end{aligned} $$

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

a <- plaplace(10,mu,lambda)
b <- plaplace(6,mu,lambda)
result3 <- a - b
result3

[1] 0.2622228

Example 6: Visualize the cumulative Laplace probability distribution

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

# create a sequence of x values
x <- seq(-5,15, by=0.02)
## Compute the Laplace pdf for each x
Fx <- plaplace(x,mu,lambda)

(f) Visualizing Laplace Distribution with plaplace() function and plot() function in R:

The cumulative probability distribution of Laplace distribution with given x, location and scale can be visualized using plot() function as follows:

## Plot the Laplace  probability dist
plot(x,Fx,type="l",xlim=c(-5,15),ylim=c(0,1),
     lwd=3, col="darkred",ylab="F(x)",
     main=expression(paste("Distribution function of L(",
        mu,"=5, ",lambda,"=2)")))

Laplace Distribution Quantiles using qlaplace() in R

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

qlaplace(p,location,scale)

where

• p : the value(s) of the probabilities,
• location : first parameter of Laplace distribution,
• scale : second parameter of Laplace distribution.

The function qlaplace(p,location,scale) gives $100*p^{th}$ quantile of Laplace distribution for given value of p, location and scale.

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

It is the inverse of plaplace() function. That is, inverse cumulative probability distribution function for Laplace distribution.

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

In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.75$. That is we need to find the $75^{th}$ quantile of given Laplace distribution.

mu <- 5
lambda <- 2
prob <- 0.75

# compute the quantile for Laplace  dist
qlaplace(0.75,mu, lambda)

[1] 6.386294

The $75^{th}$ percentile of given Laplace distribution is 6.3862944.

Visualize the quantiles of Beta Distribution

The quantiles of Beta distribution with given p, location and scale can be visualized using plot() function as follows:

p <- seq(0,1,by=0.02)
qx <- qlaplace(p,mu,lambda)
# Plot the quantiles of Laplace  dist
plot(p,qx,type="l",lwd=2,col="darkred",
     ylab="quantiles",
main=expression(paste("Quantiles of L(",
        mu,"=5, ",lambda,"=2)")))

Simulating Laplace random variable using rlaplace() function in R

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

rlaplace(n,location,scale)

where,

• n : the sample observations,
• location : first parameter of Laplace distribution,
• scale : second parameter of Laplace distribution.

The function rlaplace(n,location,scale) generates n random numbers from Laplace distribution with given location and scale.

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

In part (h), we need to generate 1000 random numbers from Laplace distribution with given $location = 5$ and $scale=2$.

(h) We can use rlaplace(1000,mu,lambda) function to generate random numbers from Laplace distribution.

## initialize sample size to generate
n <- 1000
# Simulate 1000 values From Laplace  dist
x_sim <- rlaplace(n,mu,lambda)

The below graphs shows the density of the simulated random variables from Laplace Distribution.

## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
     lwd=5,col="darkred",
     main=expression(paste("Simulated data from L(",
        mu,"=5, ",lambda,"=2)")))

If you use same function again, R will generate another set of random numbers from $L(5,2)$.

# Simulate 1000 values From Laplace  dist
x_sim_2 <- rlaplace(n,mu,lambda)

## Plot the simulated data
plot(density(x_sim_2),xlab="Simulated x",ylab="density",
     lwd=5,col="blue",
     main=expression(paste("Simulated data from L(",
        mu,"=5, ",lambda,"=2)")))

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 Laplace  dist
x_sim_3 <- rlaplace(n,mu,lambda)

## Plot the simulated data
plot(density(x_sim_3),xlab="Simulated x",ylab="density",
     lwd=5,col="darkred",
     main=expression(paste("Simulated data from L(",
        mu,"=5, ",lambda,"=2)")))

set.seed(1457)
# Simulate 1000 values From Laplace  dist
x_sim_4 <- rlaplace(n,mu,lambda)

## Plot the simulated data
plot(density(x_sim_4),xlab="Simulated x",ylab="density",
     lwd=5,col="darkred",
     main=expression(paste("Simulated data from L(",
        mu,"=5, ",lambda,"=2)")))

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

hist(x_sim_4,breaks = 30,col="red4",
     main=expression(paste("Histogram L(",
        mu,"=5, ",lambda,"=2)")))

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

Discrete Distributions Using R

Binomial distribution in R
Poisson distribution in R
Geometric distribution in R
Negative Binomial distribution in R
Hypergeometric distribution in R

Continuous Distributions Using R

Uniform distribution in R
Exponential distribution in R
Normal distribution in R
Log-normal distribution in R
Gamma distribution in R
Beta distribution in R
Cauchy distribution in R
Logistic distribution in R
Weibull distribution in R

Endnote

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

To learn more about R code for discrete and continuous probability distributions, please refer to the following tutorials:

Probability Distributions using R

Let me know in the comments below, if you have any questions on Laplace Distribution using R and your thought on this article.