Question

Below is R code (R version 3.6.2 (2019-12-12) -- "Dark and Stormy Night") to plot a pareto distribution. For part (c), I'm unable to get the density curve to show up in the histogram. Thanks!

Problem 1. The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail. The Pareto($a$,$b$) distribution has pdf

$$
f(x) = \frac{ab^{a}}{x^{a+1}}, \quad x \geq b
$$

Set the seed number

set.seed(210)

(a) Use the inverse transform method to generate a random sample of size 1000 from the Pareto(3,2) distribution.

x <- runif(1000)
inv.cdf.pareto <- 2/(1-x)^(1/3)
head(inv.cdf.pareto)
tail(inv.cdf.pareto)

(b) Plot the density histogram of the sample.

hist(inv.cdf.pareto, prob = TRUE, col="gray", xlab="x")

(c) Add the theoretical density curve $f(x)$ to the density histogoram.

fx <- (3*2^3)/(x^4)
hist(inv.cdf.pareto, prob = TRUE, col="gray", xlab="x")
lines(x,fx)

Answer 1

a) Use the inverse transform method to generate a random sample of size 1000 from the Pareto(3,2) distribution.

> set.seed(210)
> x <- runif(1000)
> inv.cdf.pareto <- 2/(1-x)^(1/3)
> head(inv.cdf.pareto)
[1] 2.540347 3.647373 4.213790 2.435244 2.250679 2.057318
> tail(inv.cdf.pareto)
[1] 2.029346 2.514305 2.151040 2.103859 4.200316 2.215587

b) Plot the density histogram of the sample.

> hist(inv.cdf.pareto, prob = TRUE, col="gray", xlab="x")

c)

> lines(density(inv.cdf.pareto))