Question

Determine a method to generate random observations for the Cauchy distribution with pdf $$ f(x)=\frac{1}{\pi\left(1+x^{2}\right)}, \quad-\infty

Short answer

The function in R that returns a random sample of observations is written as follows: \n\n ```R\ncauchy_sample <- function(n)\n{\n u <- runif(n)\n x <- tan(pi * (u - 0.5))\n return(x)\n}\n```\nThis function uses the inverse transform sampling method by taking the inverse of the CDF of the Cauchy distribution, which transforms uniformly distributed random variables into random variables of the Cauchy distribution.

Step-by-step solution

Understanding the Inverse Transform Sampling Method

The inverse transform sampling method is a technique for random number generation with an arbitrary probability distribution. It works by transforming uniformly distributed random variables, which are easy to generate, into random variables with the required distribution. In the context of the Cauchy distribution, the cumulative distribution function (CDF) is used to calculate the inverse transform.

Deriving the Inverse of Cumulative Distribution Function

For the Cauchy distribution with the given pdf, the cumulative distribution function (CDF) is \(F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan(x)\), the range of which is (0,1). Therefore, the associated inverse function is \(F^{-1}(y) = \tan(\pi(y-\frac{1}{2}))\). Using this formula, uniformly distributed random variables can be converted into the required Cauchy distribution.

Writing the R Function

Now, a function in R is created in the following way: \n\n```R\n cauchy_sample <- function(n)\n{\n u <- runif(n)\n x <- tan(pi * (u - 0.5))\n return(x)\n}\n```\nIn this function, `n` is the number of random variables to be generated, `runif(n)` generates `n` random variables which are uniformly distributed, and the `tan(pi * (u - 0.5))` converts these variables into the Cauchy distribution using the inverse CDF, as derived in previous step. The function returns the `n` number of random variables generated

Key concepts

Inverse Transform Sampling

Inverse transform sampling is a method used for generating random numbers from any probability distribution by utilizing its cumulative distribution function (CDF). When working with complex distributions like the Cauchy distribution, using widely available uniform random number generators is not enough. The process involves two main steps: generating a uniform random sample and then transforming that sample using the inverse of the CDF.

Firstly, a uniform random sample is produced, which can easily be done using a standard computer algorithm. These random numbers fall uniformly across the interval [0, 1], which means each subinterval of [0, 1] has an equal chance of being selected. Then, the inverse function of the CDF, known as the quantile function, is applied to these numbers. This transforms them so that the new values follow the desired target distribution. In the context of our exercise, the Cauchy distribution has an inverse CDF that can be expressed analytically, which makes the transform straightforward.

Probability Density Function (pdf)

The probability density function (pdf) represents how the values of a continuous random variable are distributed. It is the function that describes the likelihood of a random variable taking on a particular value. The pdf for the Cauchy distribution, given in the exercise, is \( f(x) = \frac{1}{\pi(1+x^2)} \), for all real numbers \(x\). This function illustrates the 'bell-shaped' curve, which is characteristic of the Cauchy distribution. Unlike a normal distribution, the Cauchy pdf has fatter tails, meaning it has a higher chance of producing outliers.

The pdf plays a crucial role when generating random samples, as it helps to define the shape of the CDF, which is then inverted in the transform sampling method. Because the integral of the pdf over its entire domain is equal to 1, it encompasses the entire probability space.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a real-valued random variable is the probability that the variable will take a value less than or equal to a given number. Essentially, it gives the area under the probability density function curve to the left of a certain value. For the Cauchy distribution, the CDF is \( F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan(x) \).

When utilizing the inverse transform sampling method, the CDF is especially important because its inverse is used to convert uniformly distributed samples into samples following the desired probability distribution. In this exercise, to generate random samples from the Cauchy distribution, the inverse of its CDF, which is \( F^{-1}(y) = \tan(\pi(y-\frac{1}{2})) \), is applied to uniform random samples.

R programming

R is a programming language and environment commonly used for statistical computing and graphics. It offers a variety of packages and functions for data analysis, making it a popular choice for statisticians and data scientists. The `runif` function in R generates uniformly distributed random numbers, which are a prerequisite for inverse transform sampling.

In our exercise, the R function \( cauchy_sample \) is defined to generate a random sample from the Cauchy distribution using the inverse transform sampling method. This custom function takes an argument \( n \), the sample size, and utilizes the `runif` function to create \( n \) uniform samples. Then, it transforms them using the previously derived inverse CDF of the Cauchy distribution to produce the final sample of Cauchy-distributed random numbers. Such functions are fundamental in simulating real-world phenomena and conducting statistical experiments in R.