TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW03.5

The Pareto distribution

Generating random numbers

Problem 3c shows that if $X\sim {\mathcal {U}}(0,1)$, then $Y=F^{-1}(X)$ generates a random number $Y$ from any continuous distribution with the specified cumulative distribution function $F$. This gives us the following algorithm to generate random numbers from a random variable $Y$ with the given cumulative distribution function $F$.

Algorithm: Inversion method

1. compute the inverse $F^{-1}$ of $F$
2. generate $n$ independent random numbers $x_{1},x_{2},\dots ,x_{n}$ from ${\mathcal {U}}(0,1)$
3. compute $y_{1}=F^{-1}(x_{1}),y_{2}=F^{-1}(x_{2}),\dots ,y_{n}=F^{-1}(x_{n})$.

Then, $y_{1},y_{2},\dots ,y_{n}$ are $n$ independent random observations of the random variable $Y$.

In R, generate 15 observations from a random variable $X$ with the Pareto distribution with

$\alpha =m=2$ (see Problem 4) by applying the inversion method.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

library(EnvStats)


# HW 3 Problem 5
# Generate 15 observations form a random variable X with the Pareto distribution with a = m = 2 by applying the inversion method

F <- function(x) {
  ifelse(x >= 2, 1 - (2 / x)^2, 0)
}

# lets look at the function
x <- seq(0, 20, 0.1)
plot(x, F(x))


F_inv <- function(p) {
  ifelse(p > 0, 2 / sqrt(1 - p), 0)
}

# and we take a look at its inverse
p <- seq(0, 1, 0.01)
plot(p, F_inv(p))


# now for the real algorithm:

# 1) generate 15 samples from standard uniform distribution
p <- runif(15, 0, 1)
p

# 2) map them to values from the pareto distribution
X <- F_inv(p)
X
hist(X)


# just to very: we use EnvStats to see how a histogram of those samples should look like
X_test <- qpareto(p, 2, 2)
hist(X_test)