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# 数学代写|统计计算作业代写Statistical Computing代考|Discrete distributions

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## 数学代写|统计计算作业代写Statistical Computing代考|transformations described

Building on the methods from Section 1.1, in this and the following sections we will study methods to transform an i.i.d. sequence of $\mathcal{U}[0,1]$-distributed random variables into an i.i.d. sequence from a prescribed target distribution. The methods from the previous section were inexact, since the output of a PRNG is not ‘truly random’. In contrast, the transformations described in this and the following sections can be carried out with complete mathematical rigour. We will discuss different methods for generating samples from a given distribution, applicable to different classes of target distributions. In this section we concentrate on the simplest case where the target distribution only takes finitely or countably infinitely many values.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|generate samples

As a first example, we consider the uniform distribution on the set $A=$ ${0,1, \ldots, n-1}$, denoted by $\mathcal{U}{0,1, \ldots, n-1}$. Since the set $A$ has $n$ elements, a random variable $X$ with $X \sim \mathcal{U}{0,1, \ldots, n-1}$ satisfies
$$P(X=k)=\frac{1}{n}$$
for all $k \in A$. To generate samples from such a random variable $X$, at first it may seem like a good idea to just use a PRNG with state space $A$, for example the LCG with modulus $m=n$. But considering the fact that the maximal period length of a PRNG is restricted to the size of the state space, it becomes clear that this is not a good idea. Instead we will follow the approach to first generate a continuous sample $U \sim \mathcal{U}[0,1]$ and then to transform this sample into the required discrete uniform distribution. A method to implement this idea is described in the following lemma.