Sampling from a distribution

If $U$ is uniformly distributed on $[0,1]$, what is the distribution of $X = \lfloor n U\rfloor +1$?

Given the first $n$ integers in any sequence $S_{0}$, proceed thus:

(a) pick any position $P_{0}$ from $\left\{ 1,2, \ldots , n\right\}$ at random, and swap the integer in that place of $S_{0}$ with the integer in the $n$th place of $S_{0}$, yielding $S_{1}$.

(b) pick any position $P_{1}$ from $\left\{ 1,2, \ldots , n-1\right\}$ at random, and swap the integer in that place of $S_{1}$ with the integer in the $(n-1)$ th place of $S_{1}$, yielding $S_{2}$,

(c) at the $(r-1)$ th stage the integer in position $P_{r-1}$, chosen randomly from $\left\{ 1,2, \ldots , n-r+1\right\}$, is swapped with the integer at the $(n-r+1)$ th place of the sequence $S_{r-1}$. Show that $S_{n-1}$ is equally likely to be any of the $n$ ! permutations of $\left\{ 1,2, \ldots , n\right\}$.

Use the rejection method to sample from the gamma density $\Gamma (\lambda , t)$ where $t( \geq 1)$ may not be assumed integral. [Hint: You might want to start with an exponential random variable with parameter $1 / t$.]

Show how to sample from the beta density $\beta (\alpha , \beta )$ where $\alpha , \beta \geq 1$. [Hint: Use Exercise (4.11.3).]

Describe three distinct methods of sampling from the density $f(x) = 6 x(1-x), 0 \leq x \leq 1$.

A finite real vector is called a probability vector if it has non-negative entries with sum 1. Show that a probability vector $\mathbf{p}$ of length $n$ may be written in the form

where each $\mathbf{v}_{r}$ is a probability vector with at most two non-zero entries. Describe a method, based on this observation, for sampling from $\mathbf{p}$ viewed as a probability mass function.

Let $U_{1}$ and $U_{2}$ be independent and uniformly distributed on $[0,1]$, and let $T_{i} = 2 U_{i}-1$. Show that, conditional on the event that $R = \sqrt{T_{1}^{2}+T_{2}^{2}} \leq 1$,

are independent standard normal random variables.

Let $U$ be uniform on $[0,1]$ and $0 < q < 1$. Show that $X = 1+\lfloor \log U / \log q\rfloor$ has a geometric distribution.

A point $(X, Y)$ is picked uniformly at random in the semicircle $x^{2}+y^{2} \leq 1, x \geq 0$. What is the distribution of $Z = Y / X$?

Let $X$ be a non-negative integer-valued random variable with $h(r) = \mathbb {P}\left(X = r \mid X \geq r\right)$. If $\left\{ U_{i}: i \geq 0\right\}$ are independent and uniform on $[0,1]$, show that $Z = \min \left\{ n : U_{n} \leq h(n)\right\}$ has the same distribution as $X$.

Let $g\left(x_{1}, x_{2}, \ldots , x_{n}\right)$ be an increasing function in all its variables, and let $\left\{ U_{r}: r \geq 1\right\}$ be independent and identically distributed random variables having the uniform distribution on $[0,1]$. Show that

[Hint: Use the FKG inequality of Problem (3.11.18).] Explain how this can help in the efficient estimation of $I = \int_{0}^{1} g(\mathbf{x}) d \mathbf{x}$.

We wish to estimate $I = \int g(x) f_{X}(x) d x = \mathbb {E}\left[g(X)\right]$, where either it is difficult to sample from the density $f_{X}$, or $g(X)$ has a very large variance. Let $f_{Y}$ be equivalent to $f_{X}$, which is to say that, for all $x, f_{X}(x) = 0$ if and only if $f_{Y}(x) = 0$. Let $\left\{ Y_{i}: 0 \leq i \leq n\right\}$ be independent random variables with density function $f_{Y}$, and define

A technique invented by J. M. Hammersley and K. W. Morton in 1956.

Show that:

Construct two distinct methods of sampling from the arc sine density function

Let $(X, Y)$ be a random point chosen uniformly on the unit disk. Show that

is uniformly distributed on the unit sphere.

It is desired to sample from a density function $f_{X}$ satisfying $f_{X} = f_{1}+f_{2}$ where $f_{1}, f_{2}: \mathbb {R} \rightarrow [0, \infty )$, and $f_{1} \leq f_{Y}$ for some random variable $Y$. Let $U$ be uniformly distributed on $[0,1]$, and independent of $Y$. From $Y$ and $U$, we generate another random variable $Z$ as follows. If $U > f_{1}(Y) / f_{Y}(Y)$, let $Z$ have density $f_{2}(z) / \int_{-\infty }^{\infty } f_{2}(u) d u$; otherwise set $Z = Y$. Show that $Z$ has density function $f_{X}$.

The function $F^{-1}$ defined in Theorem (4.11.1) may be called the quantile function of the random variable in question.

For an arbitrary (not necessarily continuous) distribution function $F = F_{X}$, define

Let $U$ be uniformly distributed on $(0,1)$ and independent of $X$. We call $G(X, U)$ the generalized distributional transform of $X$. Show that $W = G(X, U)$ is uniformly distributed on $(0,1)$, and that the random variable $F^{-1}(W)$ has distribution function $F$.

Let $\left(X_{1}, X_{2}, \ldots , X_{n}\right)$ be a random vector with joint distribution function $F$ and continuous marginal distribution functions $F_{1}, F_{2}, \ldots , F_{n}$. Show that there exists a 'copula-form' distribution function $G\left(u_{1}, u_{2}, \ldots , u_{n}\right)$ such that $F\left(x_{1}, x_{2}, \ldots , x_{n}\right) = G\left(F_{1}\left(x_{1}\right), F_{2}\left(x_{2}\right), \ldots , F_{n}\left(x_{n}\right)\right)$. [A 'copula-form' distribution function is one with the property that all its marginals are uniform on $[0,1]$].