Cauchy Distribution probabilities using R
In this tutorial, you will learn about how to use dcauchy(), pcauchy(), qcauchy() and rcauchy() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Cauchy distribution.
Before we discuss R functions for Cauchy distribution, let us see what is Cauchy distribution.
Cauchy Distribution
Cauchy distribution distribution is a continuous type probability distribution.
Let $X\sim C(\mu,\lambda)$. Then the probability distribution of $X$ is
$$ \begin{aligned} f(x)&= \begin{cases} \frac{\lambda}{\pi}\cdot \frac{1}{\lambda^2+(x-\mu)^2}, & -\infty < x< \infty; \\ & -\infty < \mu< \infty, \lambda>0;\\ 0, & Otherwise. \end{cases} \end{aligned} $$
where $\mu$ is the location parameter and $\lambda$ is the scale parameter of Cauchy distribution.
The distribution function of Cauchy distribution is
$$ \begin{equation*} F(x) =\frac{1}{\pi}\tan^{-1}\bigg(\frac{x-\mu}{\lambda}\bigg) + \frac{1}{2}. \end{equation*} $$
Read more about the theory and results of Cauchy distribution here.
Cauchy probabilities using dcauchy() 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 Cauchy distribution using R is
dcauchy(x,location, scale)where
x: the value(s) of the variable and,location: location parameter of Cauchy distribution,scale: scale parameter of Cauchy distribution.
The dcauchy() 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 Cauchy distribution).
Numerical Problem for Cauchy Distribution
To understand the four functions dcauchy(), pcauchy(), qcauchy() and rcauchy(), let us take the following numerical problem.
Cauchy Distribution Example
Let random variable $X$ follows a Cauchy distribution with $\mu=2$ and $\lambda=4$.
(a) Find the value of the density function at $x=2.5$.
(b) Plot the graph of Cauchy probability distribution.
(c) Find the probability that, $X$ is less than 3.
(d) Find the probability that, $X$ is greater than 4.
(e) Find the probability that, $X$ is less than 1 but greater than 3.5.
(f) Plot the graph of cumulative Cauchy probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.80$?
(h) Simulate 1000 Cauchy distributed random variables with $\mu= 2$ and $\lambda = 4$.
Let $X$ denote the daily proportion of major automobile accidents across the United States. Given that $X\sim C(\mu=2, \lambda=4)$.
Example 1: How to use dcauchy() function in R?
To find the value of the density function at $x=2.5$ we need to use dcauchy() function.
Let $X\sim C(2,4)$.
First let us define the given parameters as
# parameter 1 location
mu <- 2
# parameter 2 scale
lambda <- 4The probability density function of $X$ is
$$ \begin{aligned} f(x)&= \frac{4}{\pi}\cdot \frac{1}{4^2+(x-2)^2},\\ &\quad\text{for } -\infty \leq x \leq \infty. \end{aligned} $$
For part (a), we need to find the density function at $x=2.5$. That is $f(2.5)$.
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 dcauchy() function in R.
(a) The value of the density function at $x=2.5$ is
$$ \begin{aligned} f(2.5)&=\frac{4}{\pi}\cdot \frac{1}{4^2+(2.5-2)^2}\\ &=\frac{4}{\pi}\cdot \frac{1}{4^2+(2.5-2)^2} &= 0.0783532 \end{aligned} $$
The above probability can be calculated using dcauchy(2.5,2,4) function in R.
# Compute Cauchy probability
result1 <- dcauchy(2.5,mu,lambda)
result1[1] 0.0783532Example 2 Visualize Cauchy probability distribution
Using dcauchy() function we can compute Cauchy distribution probabilities for given x, location and scale. To plot the probability density function of Cauchy distribution, we need to create a sequence of x values and compute the corresponding probabilities.
# create a sequence of x values
x <- seq(-10,14, by=0.02)
## Compute the Cauchy pdf for each x
px<-dcauchy(x,mu,lambda)(b) Visualizing Cauchy Distribution with dcauchy() function and plot() function in R:
The probability density function of Cauchy distribution with given 2 and 4 can be visualized using plot() function as follows:
## Plot the Cauchy probability dist
plot(x,px,type="l",xlim=c(-10,14),ylim=c(0,max(px)),
lwd=3, col="darkred",ylab="f(x)",
main=expression(paste("PDF of C(",
mu,"=2, ",lambda,"=4)")))
abline(v=c(2),lty=2,col="gray")
Cauchy cumulative probability using pcauchy() function in R
The syntax to compute the cumulative probability distribution function (CDF) for Cauchy distribution using R is
pcauchy(q,location, scale)where
q: the value(s) of the variable,location: first parameter of Cauchy distribution,scale: second parameter of Cauchy distribution.
Using this function one can calculate the cumulative distribution function of Cauchy distribution for given value(s) of q (value of the variable x), location and scale.
Example 3: How to use pcauchy() function in R?
In the above example, for part (c), we need to find the probability $P(X\leq 3)$.
(c) The probability that $X$ is less than $3$ is
$$ \begin{aligned} P(X \leq 3) &=F(3)\\ &=0.5+\frac{1}{\pi} tan^{-1}\big(\frac{3-2}{4}\big)\\ &=0.5 + \frac{1}{3.1416}tan^{-1}\big(0.25\big)\\ &=0.5 + \frac{1}{3.1416}(0.245)\\ &= 0.578 \end{aligned} $$
## Compute cumulative Cauchy probability
result2 <- pcauchy(3,mu,lambda)
result2[1] 0.5779791Example 4: How to use pcauchy() function in R?
In the above example, for part (d), we need to find the probability $P(X \geq 4)$.
(d) The probability that $X$ is greater than $4$ is
$$ \begin{aligned} P(X > 4) &=1- P(X < 4)\\ &= 1- F(4)\\ &=1-\bigg(0.5+\frac{1}{\pi} tan^{-1}\big(\frac{4-2}{4}\big)\bigg)\\ &=0.5 - \frac{1}{3.1416}tan^{-1}\big(0.5\big)\\ &=0.5 - \frac{1}{3.1416}(0.4636)\\ &= 0.3524 \end{aligned} $$
To calculate the probability that a random variable $X$ is greater than a given number one can use the option lower.tail=FALSE in pcauchy() function.
Above probability can be calculated easily using pcauchy() function with argument lower.tail=FALSE as
$P(X \geq 4)=$ pcauchy(4,mu,lambda,lower.tail=FALSE)
or by using complementary event as
$P(X \geq 4) = 1- P(X\leq 4)$= 1- pcauchy(1,mu,lambda)
# compute cumulative Cauchy probabilities
# with lower.tail False
pcauchy(4,mu,lambda,lower.tail=FALSE)[1] 0.3524164# Using complementary event
1-pcauchy(4,mu,lambda)[1] 0.3524164Example 5: How to use pcauchy() function in R?
One can also use pcauchy() function to calculate the probability that the random variable $X$ is between two values.
(e) The probability that $X$ is between $1$ and $3.5$ is
$$ \begin{aligned} P(1 \leq X \leq 3.5)&=P(X\leq 3.5)-P(X\leq 1)\\ &=F(3.5) -F(1)\\ &=\bigg[0.5+\frac{1}{\pi} tan^{-1}\big(\frac{3.5-2}{4}\big)\bigg]-\bigg[0.5+\frac{1}{\pi} tan^{-1}\big(\frac{1-2}{4}\big)\bigg]\\ &=\frac{1}{\pi} tan^{-1}\big(0.375\big)-\frac{1}{\pi} tan^{-1}\big(-0.25\big)\\ &=\frac{1}{3.1416}(0.3588)-\frac{1}{3.1416}(-0.245)\\ &=0.1922 \end{aligned} $$
The above probability can be calculated using pcauchy() function as follows:
a <- pcauchy(3.5,mu,lambda)
b <- pcauchy(1,mu,lambda)
result3 <- a - b
result3[1] 0.1921794Example 6: Visualize the cumulative Cauchy probability distribution
Using pcauchy() function we can compute Cauchy cumulative probabilities (CDF) for given x, location and scale. To plot the CDF of Cauchy distribution, we need to create a sequence of x values and compute the corresponding cumulative probabilities.
# create a sequence of x values
x <- seq(-10,14, by=0.02)
## Compute the Cauchy pdf for each x
Fx <- pcauchy(x,mu,lambda)(f) Visualizing Cauchy Distribution with pcauchy() function and plot() function in R:
The cumulative probability distribution of Cauchy distribution with given x, location and scale can be visualized using plot() function as follows:
## Plot the Cauchy probability dist
plot(x,Fx,type="l",xlim=c(-10,14),ylim=c(0,1),
lwd=3, col="darkred",ylab="F(x)",
main=expression(paste("Distribution function of C(",
mu,"=2, ",lambda,"=4)")))
Cauchy Distribution Quantiles using qcauchy() in R
The syntax to compute the quantiles of Cauchy distribution using R is
qcauchy(p,location,scale)where
p: the value(s) of the probabilities,location: first parameter of Cauchy distribution,scale: second parameter of Cauchy distribution.
The function qcauchy(p,location,scale) gives $100*p^{th}$ quantile of Cauchy distribution for given value of p, location and scale.
The $p^{th}$ quantile is the smallest value of Cauchy random variable $X$ such that $P(X\leq x) \geq p$.
It is the inverse of pcauchy() function. That is, inverse cumulative probability distribution function for Cauchy distribution.
Example 7: How to use qcauchy() function in R?
In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.80$. That is we need to find the $80^{th}$ quantile of given Cauchy distribution.
mu <- 2
lambda <- 4
prob <- 0.80# compute the quantile for Cauchy dist
qcauchy(0.80,mu, lambda)[1] 7.505528The $85^{th}$ percentile of given Cauchy distribution is 9.850442.
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 <- qcauchy(p,mu,lambda)
# Plot the quantiles of Cauchy dist
plot(p,qx,type="l",lwd=2,col="darkred",
ylab="quantiles",
main=expression(paste("Quantiles of C(",
mu,"=2, ",lambda,"=4)")))
Simulating Cauchy random variable using rcauchy() function in R
The general R function to generate random numbers from Cauchy distribution is
rcauchy(n,location,scale)where,
n: the sample observations,location: first parameter of Cauchy distribution,scale: second parameter of Cauchy distribution.
The function rcauchy(n,location,scale) generates n random numbers from Cauchy distribution with given location and scale.
Example 8: How to use rcauchy() function in R?
In part (h), we need to generate 1000 random numbers from Cauchy distribution with given $location = 2$ and $scale=4$.
(h) We can use rcauchy(1000,mu,lambda) function to generate random numbers from Cauchy distribution.
# location parameter
mu <- 2
# scale parameter
lambda <- 4
## initialize sample size to generate
n <- 10000
# Simulate 10000 values From Cauchy dist
x_sim <- rcauchy(n,mu,lambda)
summary(x_sim) Min. 1st Qu. Median Mean 3rd Qu. Max.
-2427.697 -2.085 1.984 1.600 6.045 3567.315 The below graphs shows the density of the simulated random variables from Cauchy Distribution.
## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
xlim=c(-200,200),lwd=5,col="darkred",
main=expression(paste("Simulated Data from C(",
mu,"=2, ",lambda,"=4)")))
The histogram of simulated random variables from Cauchy distribution is as follows:
hist(x_sim,breaks = 1000,xlim=c(-200,200),col="red4",
main=expression(paste("Histogram C(",
mu,"=2, ",lambda,"=4)")))
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
Laplace 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 Cauchy distribution in R programming. You also learned about how to simulate a Cauchy 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 Cauchy Distribution using R and your thought on this article.
