Chapter 1 - Generating random variables - Exercises

Exercise 1.1 (inversion method) 

Prove the above generation schemes.

1. Exponential distribution. Let $\lambda>0$. Then $$X=-\frac{1}{\lambda}\log(U)$$ has an exponential distribution with parameter $\lambda$ (denoted  by ${\cal E}xp(\lambda)$) which has the density $\lambda e^{-\lambda x}\mathbb{1}_{x\geq 0}$.
2. Geometric distribution. The random variable $$X\overset d= 1+\lfloor Y \rfloor$$where $Y\overset d= {\cal E}xp(\lambda)$ has a geometric distribution with parameter $p$ such that $\lambda=-\log(1-p)$ ($\mathbb{P}(X=n)=p(1-p)^{n-1}$ for $n\geq 1$).
3. Cauchy distribution. Let $\sigma>0$. Then $$X=\sigma \tan\big(\pi(U-\frac 12)\big)$$ is a Cauchy random variable with parameter $\sigma$, whose density is $\frac{ \sigma }{\pi(x^2+\sigma^2) }\mathbb{1}_{x\in \mathbb{R}}$.
4. Rayleigh distribution. Let  $\sigma>0$. Then $$X=\sigma \sqrt{-2\log U}$$ is a Rayleigh random variable with parameter $\sigma$, whose density is $\frac{ x }{\sigma^2 }e^{-\frac{ x^2 }{2\sigma^2 }}\mathbb{1}_{x\geq 0}$.
5. Pareto distribution. Let $(a, b)\in]0,+\infty[^2$. Then $$X=\frac{ b }{U^{\frac{ 1 }{a }} }$$ is a Pareto random variable with parameters $(a,b)$, whose density is  $\frac{a b^a}{x^{a+1}}\mathbb{1}_{x\geq b }$.
6. Weibull distribution. Let $(a, b)\in]0,+\infty[^2$. Then $$X=b(-\log U)^{\frac{ 1 }{a }}$$ is a Weibull random variable with parameters $(a,b)$, whose density is $\frac{a}{ b^a}x^{a-1}e^{-(x/b)^a}\mathbb{1}_{x\geq 0}$.
7. Triangular distribution. $(1 - \sqrt{U})$ with $U\overset{d}= {\cal U}([0,1])$ has the triangular distribution on $[0,1]$ (with density $2(1 -x)\mathbb{1}_{[0,1]}$).

Exercise 1.2 (Box-Muller transform)
Let $X$ and $Y$ be two independent standard Gaussian random variables. Define $(R,\theta)$ as the polar coordinates of $(X,Y)$:
$$X=R\cos(\theta), \quad Y=R\sin(\theta)$$ with $R\geq 0$ and $\theta\in[0,2\pi[$.
Prove that $R^2$ and $\theta$ are two independent random variables, the first one has the distribution of ${\cal E}xp(\frac 12)$, the second one is uniformly distributed on $[0,2\pi]$.
Solution.

Exercise 1.3 (acceptance-rejection method)
Proposition I.3.2 serves to design the acceptance-rejection method and it is stated as follows. Let $X$ and $Y$ be two random variables with values in $\mathbb{R}^d$, whose densities with respect to a reference measure $\mu$ are $f$ and $g$ respectively. Suppose that there exists a constant $c(\geq 1)$ satisfying $c\;g(x)\geq f(x)\quad \mu-\mbox{a.e.}$. Let $U$ be a random variable uniformly distributed on $[0,1]$ and independent of $Y$: then, the distribution of  $Y$ given
$\{c\; U \; g(Y)< f(Y) \}$ is the distribution $X$.
Here we study a variant of the above result. Let $c>0$. Show the following statements.

1. Let $Y$ be a $d$-dimensional random variable with density $g$ and let $U\overset{d}= {\cal U}([0,1])$ independent of $Y$. Then, $(Y, cUg(Y))$ is a random vector uniformly distributed on $$A_{cg} = \{(x,z) \in {\mathbb R}^{d}\times {\mathbb R}: 0 \leq z \leq cg(x) \}.$$ 
2. Conversely, if $(Y,Z)$ is uniformly distributed on $A_{cg}$, then the distribution of $Y$ has a density equal to $g$. 

From the above, deduce another proof of Proposition I.3.2 when the reference measure $\mu$ is the Lebesgue measure.

Exercise 1.4 (acceptance-rejection method)
Show that the following algorithm generates a standard Gaussian random variable.
  $x$: double;
  $u$: double;
  Repeat
     $x \leftarrow$ simulation according to ${\cal E}xp(1)$;
     $u \leftarrow$ simulation according to ${\cal U}([-1,1])$, independent of $x$;
  Until $(x-1)^2\leq -2 \times \log(|u|)$
  Return $x$ if $u>0$ and $-x$ otherwise
Solution.

Exercise 1.5 (acceptance-rejection method)
What is the output distribution of the following algorithm?

  $u$: double; 

  $v$: double; 

  Repeat

     $u \leftarrow$ simulation according to ${\cal U}([-1,1])$;

     $v \leftarrow$ simulation according to ${\cal U}([-1,1])$, independent of $u$;

  Until $(1+v^2)\times |u|\leq 1$

  Return $v$ if $u>0$ and $1/v$ otherwise

Exercise 1.6 (ratio-of-uniforms method, Gamma distribution)
Using the ratio-of-uniforms method, design an algorithm for simulating the Gamma distribution $\Gamma(a,\theta)$ ($a\geq 1,\theta>0$) which density is $$p_{a,\theta}(z)=\frac{\theta^a z^{a-1}}{\Gamma(a)}e^{-\theta z}1_{\{z>0\}}.$$ Hint: first reduce to the case $\theta=1$.
Solution.

Exercise 1.7 (ratio-of-uniforms method)
Generalize Lemma 1.3.6 to the multidimensional case. Make the algorithm explicit in the case of the two-dimensional density $p(x,y)$ proportional to $$f(x,y)=(1+x^2+2 y^2)^{-4/3}.$$

Exercise 1.8 (Gaussian copula)
Write a simulation program for generating a bi-dimensional vector with Laplace marginals and Gaussian copula (like for Figure 1.2).

Exercise 1.9 (Archimedean copula)
Let $C$ be the Archimedean copula
$$C(u_1,...,u_d)=\phi^{-1}(\phi(u_1)+...+\phi(u_d))$$
associated with the random variable $Y$ (with the Laplace transform $\phi^{-1}(u)=\mathbb{E}(e^{-uY})$), and suppose that $Y>0$ a.s. Let $(X_i)_{1\leq i\leq d}$ be independent random variables with the uniform distribution $[0,1]$ and $Y$ be a random variable independent of $(X_i)_i$. Define
$$U_i=\phi^{-1}\Big(-\frac 1 Y \log (X_i)\Big).$$
Prove that the vector $(U_1,\dots,U_d)$ has uniform marginal distributions and that its copula is $C$.

Solution.

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