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# 数学代写|统计计算作业代写Statistical Computing代考|The inverse transform method

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## 数学代写|统计计算作业代写Statistical Computing代考|generated algorithm

The inverse transform method is a method which can be applied when the target distribution is one-dimensional, that is to generate samples from a prescribed target

Figure $1.2$ Illustration of the inverse $F^{-1}$ of a CDF $F$. At level u the function $F$ is continuous and injective; here $F^{-1}$ coincides with the usual inverse of a function. The value $v$ falls in the middle of a jump of $F$ and thus has no preimage; $F^{-1}(v)$ is the preimage of the right-hand limit of $F$ and $F\left(F^{-1}(v)\right) \neq v$. At level $w$ the function $F$ is not injective, several points map to $w$; the preimage $F^{-1}(w)$ is the left-most of these points and we have, for example, $F^{-1}(F(a)) \neq a$.
distribution on the real numbers $\mathbb{R}$. The method uses the cumulative distribution function (CDF) (see Section A.1) to specify the target distribution and can be applied for distributions which have no density.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|distribution function

Let $F$ be a distribution function. Then the inverse of $F$ is defined by
$$F^{-1}(u)=\inf {x \in \mathbb{R} \mid F(x) \geq u}$$
for all $u \in(0,1)$.
The definition of the inverse of a distribution function is illustrated in Figure 1.2. In the case where $F$ is bijective, that is when $F$ is strictly monotonically increasing and has no jumps, $F^{-1}$ is just the usual inverse of a function. In this case we can find $F^{-1}(u)$ by solving the equation $F(x)=u$ for $x$. The following algorithm can be used to generate samples from a given distribution, whenever the inverse $F^{-1}$ of the distribution function can be determined.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|DISTRIBUTION FUNCTION

F−1(你)=信息X∈R∣F(X)≥你