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# 数学代写|统计计算作业代写Statistical Computing代考|Transformation of random variables

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

Samples from a wide variety of distributions can be generated by considering deterministic transformations of random variables. The inverse transform method, introduced in Section 1.3, is a special case of this technique where we transform a uniformly distributed random variable using the inverse of a CDF. In this section, we consider more general transformations.

The fundamental question we have to answer in order to generate samples by transforming a random variable is the following: if $X$ is a random variable with values in $\mathbb{R}^{d}$ and a given distribution, and if $\varphi: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ is a function, what is the distribution of $\varphi(X)$ ? This question is answered in the following theorem.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Envelope rejection sampling

Theorem $1.34$ (transformation of random variables) Let $A, B \subseteq \mathbb{R}^{d}$ be open sets, $\varphi: A \rightarrow B$ be bijective and differentiable with continuous partial derivatives, and

let $X$ be a random variable with values in $A$. Furthermore let $g: B \rightarrow[0, \infty)$ be a probability density and define $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ by
$$f(x)= \begin{cases}g(\varphi(x)) \cdot|\operatorname{det} D \varphi(x)| & \text { if } x \in A \text { and } \ 0 & \text { otherwise }\end{cases}$$
Then $f$ is a probability density and the random variable $X$ has density $f$ if and only if $\varphi(X)$ has density $g$.

The matrix $D \varphi$ used in the theorem is the Jacobian of $\varphi$, as given in the following definition.

Definition 1.35 Let $\varphi: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ be differentiable. Then the Jacobian matrix $D \varphi$ is the $d \times d$ matrix consisting of the partial derivatives of $\varphi:$ for $i, j=1,2, \ldots, d$ we have $D \varphi(x){i j}=\frac{\partial \varphi{i}}{\partial x_{j}}(x)$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|ENVELOPE REJECTION SAMPLING

F(X)={G(披(X))⋅|这⁡D披(X)| 如果 X∈一种 和  0 否则