Question

Determine a method to generate random observations for the Cauchy distribution with pdf:

f(x) = 1/ (x²+ 1) < -[infinity] < x < [infinity] (4.8.15) ,

Write an R function that returns a random sample of observation from a Cauchy distribution. Then simulate 1000 samples from Cauchy and plot its kernel density function. On the same figure, plot the pdf of Cauchy. Comment on what you observe.

Answer 1

To generate random observations from the Cauchy distribution, use the inverse transform sampling method. Create an R function that returns a random sample of observations from the Cauchy distribution and simulate 1000 samples. Plot the kernel density function and compare it with the actual probability density function of the Cauchy distribution, observing the heavy tails and lack of concentration of probability around the mean.

Step-by-step explanation:

To generate random observations from the Cauchy distribution with the given probability density function (pdf), we can make use of the inverse transform sampling method. Follow the steps below to create an R function that returns a random sample of observations from a Cauchy distribution:

1. Write a function called cauchy_sample that takes the number of samples as input.
2. Inside the function, generate a uniform random sample using the runif() function in R.
3. Apply the inverse of the cumulative distribution function (cdf) of the Cauchy distribution to the uniform random sample to obtain the desired random sample.
4. Return the random sample.

To simulate 1000 samples from the Cauchy distribution and plot its kernel density function (KDF) along with the actual probability density function (pdf) of the Cauchy distribution, follow these steps:

1. Call the cauchy_sample function with 1000 as the number of samples.
2. Use the density() function in R to estimate the kernel density of the generated samples.
3. Plot the estimated kernel density function using the plot() function.
4. Plot the actual probability density function of the Cauchy distribution using the curve() function.
5. Add a legend to the plot to differentiate between the estimated kernel density function and the actual probability density function.

Observations:

When comparing the estimated kernel density function with the actual probability density function of the Cauchy distribution, you might observe that the estimated density function closely resembles the actual density function. The Cauchy distribution has heavy tails, which results in the estimated kernel density function having a larger spread compared to other distributions like the Gaussian distribution. Additionally, the Cauchy distribution does not have a finite mean or variance, so you might observe a lack of concentration of probability around the mean.