Question

The Cauchy(µ,σ) distribution has the pdf f(x)= 1 σ π 1 1+((x−µ)/σ)2,x ∈R.

a. Write an R function that generates a random sample of size n from a Cauchy(µ,σ)distribution. Do not use the rcauchy function. Your function should use the inverse transformation from to generate a random sample from a standard Cauchy distribution, then transform the generated sample appropriately according to the location and scale. Hint: The R function for the tangent is tan(x).

b. Use your function from part (a) to generate a random sample of size n = 10000 from a Cauchy(µ = 10,σ = 3) distribution. Give the corresponding density histogram and overly the density of a Cauchy(µ = 10,σ = 3) distribution. You can use the function dcauchy to produce the curve of the pdf.

Answer 1

R Code for the tanx function will be as follow:

n <- 10000
u <- runif(n)
c.samp <- sapply(u, function(u) tan(u))
hist(c.samp, breaks = 90, col = "blue",
main = "Hist of Cauchy")

The output graph after running the code will be as follow:

For the given question, PDF is not lear for first part hence i am removing that part & creating the PDF as follow:

F(x) = ((x-u)/)^2

In case of any change the F(x) function will change & rest will remain the same. u & are given in the question which can be used.

n <- 10000
x <- runif(n)
c.samp <- sapply(x, function(x) ((x-10)/3)^2)
hist(c.samp, breaks = 90, col = "blue",
     main = "Hist of Cauchy")

The output graph will be as follow: