Using Universality of the Uniform to simulate a Pareto distribution with parameter 1/2

Question

Problem: Use Universality of the Uniform to simulate a Pareto distribution with parameter $1/2$ in R with $10000$ simulations.

The CDF is

$$ F(x)=\left\{\begin{array}{ll} 1-x^{-1 / 2} & \text { for } x>1 \\ 0 & \text { else } \end{array}\right. $$

I think I'm suppose to use this theorem:

Theorem 5.3 .1 (Universality of the Uniform). Let $F$ be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that the inverse function $F^{-1}$ exists, as a function from $((0,1))$ to $\mathbb{R}$. We then have the following results.

1. Let $U \sim \operatorname{Unif}(0,1)$ and $X=F^{-1}(U)$. Then $X$ is an r.v. with CDF $F$.
2. Let $X$ be an r.v. with CDF $F$. Then $F(X) \sim \operatorname{Unif}(0,1)$.

I must admit that I do not understand the theorem enough to apply it this problem.

I'm stuck here.

I'm suppose to use R for this. I should make $10000$ simulations.

Answer 1

if $X\sim pareto(\alpha ,x_m)$ density_function $$F(x)=1-(\frac{x_m}{x})^{\alpha} \hspace{.5cm} x>x_m$$ so

$$X=\frac{x_m}{(1-U)^{(1/\alpha)}}$$

Now generate

sim<-10000
set.seed(100)        #optional
U <- runif(sim)
xm<-1
alpha<-1/2
X <- xm/(1-U)^(1/alpha)  
min(X)                             #[1] 1.000398      #optional
(mean(log(X)) - log(min(X)))^(-1)  #[1] 0.4981526    #optional