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# 统计代写|Transformations of Random Vectors 抽样理论代考

## 统计代写

3 Transformations of Random Vectors
A consequence of multivariate calculus is the following formula for the density of a transformed random vector:

• Let $(X, Y)$ be a random vector with density $f$. Let $(U, V)$ be such that
$$U=g_{1}(X, Y) \text { and } V=g_{2}(X, Y) .$$
3 Transformations of Random Vectors Assume that the transformation $(x, y) \longrightarrow\left(g_{1}(x, y), g_{2}(x, y)\right)$ is one to one with inverse
$$X=h_{1}(U, V) \text { and } Y=h_{2}(U, V)$$
Then the density of the transformed random vector $(U, V)$ is
$$f\left(h_{1}(u, v), h_{2}(u, v)\right)|J(u, v)|$$
where the Jacobian $J(u, v)$ is the following determinant:
$$\left|\begin{array}{ll} \partial h_{1} / \partial u & \partial h_{1} / \partial v \ \partial h_{2} / \partial u & \partial h_{2} / \partial v \end{array}\right|$$
We now use the preceding formula on an example.
Example 6 Let $X$ and $Y$ be two independent standard normal distributions. Let $U=$ $X / Y$ and $V=X$. What is the joint density of $(U, V)$ ?

Let $u=x / y$ and $v=x$. Then, solving in $x$ and $y$ we get $x=v$ and $y=v / u$. Hence, $(x, y) \longrightarrow(u, v)$ is a one to one transformation from $\mathbb{R} \times \mathbb{R}^{}$ to $\mathbb{R}^{} \times \mathbb{R}$ where $\mathbb{R}^{*}$ is the set of all real numbers except 0 . Therefore, the support of $(U, V)$ is all $(u, v)$ where $u \neq 0$.
We now compute the Jacobian
$$J(u, v)=\left|\begin{array}{cc} 0 & 1 \ -v / u^{2} & 1 / u \end{array}\right|=v / u^{2}$$
Since we assume that $X$ and $Y$ are independent standard normal distributions,
$$f(x, y)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2}$$
Therefore, the joint density of $(U, V)$ is for all $u \neq 0$ and all $v$,
$$f(u, v)=\frac{1}{2 \pi} e^{-v^{2} / 2} e^{-v^{2} /\left(2 u^{2}\right)}|J(u, v)|=\frac{1}{2 \pi} \exp \left(\frac{-v^{2}}{2}\left(1+\frac{1}{u^{2}}\right)\right) \frac{|v|}{u^{2}}$$
We now use this joint density to get the marginal density of $U$. We integrate the density above in $v$ to get
$$f_{U}(u)=\int_{-\infty}^{\infty} \frac{1}{2 \pi} \exp \left(\frac{-v^{2}}{2}\left(1+\frac{1}{u^{2}}\right)\right) \frac{|v|}{u^{2}} d v$$

3 随机向量的变换

• 令 $(X, Y)$ 为密度为 $f$ 的随机向量。让$(U, V)$ 是这样的
$$U=g_{1}(X, Y) \text { 和 } V=g_{2}(X, Y) 。$$
3 随机向量的变换假设变换$(x, y) \longrightarrow\left(g_{1}(x, y), g_{2}(x, y)\right)$ 是一对一的逆
$$X=h_{1}(U, V) \text { 和 } Y=h_{2}(U, V)$$
那么变换后的随机向量$(U, V)$的密度为
$$f\left(h_{1}(u, v), h_{2}(u, v)\right)|J(u, v)|$$
其中雅可比 $J(u, v)$ 是以下行列式：
$$\left|\begin{数组}{ll} \partial h_{1} / \partial u & \partial h_{1} / \partial v \ \partial h_{2} / \partial u & \partial h_{2} / \partial v \end{数组}\right|$$
我们现在在一个例子中使用前面的公式。