# Cauchy Density in R (4 Examples) | dcauchy, pcauchy, qcauchy & rcauchy Functions

In this R tutorial you’ll learn how to **apply the cauchy functions**.

The page contains four examples for the application of dcauchy, pcauchy, qcauchy, and rcauchy. More precisely, the article will consist of this content:

- Example 1: Cauchy Density in R (dcauchy Function)
- Example 2: Cauchy Cumulative Distribution Function (pcauchy Function)
- Example 3: Cauchy Quantile Function (qcauchy Function)
- Example 4: Random Number Generation (rcauchy Function)
- Video, Further Resources & Summary

Let’s get started:

## Example 1: Cauchy Density in R (dcauchy Function)

In Example 1, I’ll show you how to create a density plot of the cauchy distribution in R. First, we need to create an input vector containing quantiles:

x_dcauchy <- seq(0, 1, by = 0.02) # Specify x-values for cauchy function |

x_dcauchy <- seq(0, 1, by = 0.02) # Specify x-values for cauchy function

Now, we can apply the dcauchy R function to return the values of a cauchy density. In the examples of this tutorial, we use a scale of 5. However, you could modify the R syntax according to your specific preferences.

y_dcauchy <- dcauchy(x_dcauchy, scale = 5) # Apply dcauchy function |

y_dcauchy <- dcauchy(x_dcauchy, scale = 5) # Apply dcauchy function

Our cauchy density values are now stored in the data object y_dcauchy. If we want to draw a density plot based on these values, we can use the plot function as shown below:

plot(y_dcauchy) # Plot dcauchy values |

plot(y_dcauchy) # Plot dcauchy values

**Figure 1: Cauchy Density in R.**

## Example 2: Cauchy Cumulative Distribution Function (pcauchy Function)

Example 2 shows how to draw a plot of the cumulative distribution function (CDF) of the cauchy distribution. As a first step, we need to create a vector of quantiles:

x_pcauchy <- seq(0, 1, by = 0.02) # Specify x-values for pcauchy function |

x_pcauchy <- seq(0, 1, by = 0.02) # Specify x-values for pcauchy function

We can now apply the pcauchy R function to get the cauchy CDF values of our input vector:

y_pcauchy <- pcauchy(x_pcauchy, scale = 5) # Apply pcauchy function |

y_pcauchy <- pcauchy(x_pcauchy, scale = 5) # Apply pcauchy function

The final graphic can be created as follows:

plot(y_pcauchy) # Plot pcauchy values |

plot(y_pcauchy) # Plot pcauchy values

**Figure 2: Cumulative Distribution Function of Cauchy Distribution.**

## Example 3: Cauchy Quantile Function (qcauchy Function)

The qcauchy command takes an input vector of probabilities and returns the corresponding values of the cauchy quantile function. Consider the following input vector:

x_qcauchy <- seq(0, 1, by = 0.02) # Specify x-values for qcauchy function |

x_qcauchy <- seq(0, 1, by = 0.02) # Specify x-values for qcauchy function

Now, the qcauchy function is applied as follows…

y_qcauchy <- qcauchy(x_qcauchy, scale = 5) # Apply qcauchy function |

y_qcauchy <- qcauchy(x_qcauchy, scale = 5) # Apply qcauchy function

…and the corresponding scatterplot is created as follows:

plot(y_qcauchy) # Plot qcauchy values |

plot(y_qcauchy) # Plot qcauchy values

**Figure 3: Quantile Function of Cauchy Distribution.**

## Example 4: Random Number Generation (rcauchy Function)

We can also simulate random numbers that are distributed as the cauchy density. First, we need to specify a seed to ensure reproducibility and a sample size that we want to generate:

set.seed(13579) # Set seed for reproducibility N <- 10000 # Specify sample size |

set.seed(13579) # Set seed for reproducibility N <- 10000 # Specify sample size

Now, we can apply the rcauchy function to create a set of random values as follows:

y_rcauchy <- rcauchy(N, scale = 5) # Draw N cauchy distributed values y_rcauchy # Print values to RStudio console |

y_rcauchy <- rcauchy(N, scale = 5) # Draw N cauchy distributed values y_rcauchy # Print values to RStudio console

A histogram of our random data can be created as follows:

hist(y_rcauchy, # Plot of randomly drawn cauchy density xlim = c(- 200, 200), breaks = 10000, main = "") |

hist(y_rcauchy, # Plot of randomly drawn cauchy density xlim = c(- 200, 200), breaks = 10000, main = "")

**Figure 4: Random Sample with Cauchy Distribution.**

## Video, Further Resources & Summary

In case you need more explanations on the R programming codes of this page, you may watch the following video of my YouTube channel. I’m explaining the examples of this tutorial in the video:

*The YouTube video will be added soon.*

You might also have a look at the other articles on probability distributions and the simulation of random numbers in the R programming language:

- Bernoulli Distribution in R
- Beta Distribution in R
- Binomial Distribution in R
- Bivariate & Multivariate Distributions in R
- Cauchy Distribution in R
- Chi-Squred Distribution in R
- Exponential Distribution in R
- F Distribution in R
- Gamma Distribution in R
- Geometric Distribution in R
- Hypergeometric Distribution in R
- Log Normal Distribution in R
- Logistic Distribution in R
- Negative Binomial Distribution in R
- Normal Distribution in R
- Poisson Distribution in R
- Student t Distribution in R
- Studentized Range Distribution in R
- Uniform Distribution in R
- Weibull Distribution in R
- Wilcoxon Signedank Statistic Distribution in R
- Wilcoxonank Sum Statistic Distribution in R

Besides the video, you may want to have a look at the other tutorials of my website. I have published several posts already:

You learned in this tutorial how to **handle the cauchy functions** in the R programming language. Don’t hesitate to let me know in the comments section below, in case you have any additional comments or questions.

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