二重积分换元法

雅可比行列式

现有函数

f_1(x_1,x_2,x_3,\cdots,x_n)\\f_2(x_1,x_2,x_3,\cdots,x_n)\\f_3(x_1,x_2,x_3,\cdots,x_n)\\\cdots\\f_m(x_1,x_2,x_3,\cdots,x_n)

那么雅可比行列式为:

J=\left|\begin{array}{cc}
 \dfrac{\partial f_1}{\partial x_1}&\dfrac{\partial f_1}{\partial x_2}&\dfrac{\partial f_1}{\partial x_3}&\cdots&\dfrac{\partial f_1}{\partial x_n}\\
 \dfrac{\partial f_2}{\partial x_1}&\dfrac{\partial f_2}{\partial x_2}&\dfrac{\partial f_2}{\partial x_3}&\cdots&\dfrac{\partial f_2}{\partial x_n}\\
 \vdots&\vdots&\vdots&\ddots&\vdots\\
 \dfrac{\partial f_m}{\partial x_1}&\dfrac{\partial f_m}{\partial x_2}&\dfrac{\partial f_m}{\partial x_3}&\cdots&\dfrac{\partial f_m}{\partial x_n}\\
\end{array}\right|

换元法

定积分换元法 二重积分换元法
\int_{a}^{b} f(x) \mathrm{d} x \xlongequal{x=\varphi(t)} \int_{\alpha}^{\beta} f[\varphi(t)] \varphi^{\prime}(t) \mathrm{d} t \iint_{D x y} f(x, y) \mathrm{d} x \mathrm{~d} y \xlongequal[y=y(u, v)]{x=x(u, v)}\iint_{D_{u v}} f[x(u, v), y(u, v)]\left\vert\frac{\partial(x, y)}{\partial(u, v)}\right\vert \mathrm{d} u \mathrm{~d} v

a. f(x) \rightarrow f[\varphi(t)]
b. \int_{a}^{b} \rightarrow \int_{\alpha}^{\beta}
c. \mathrm{d} x \rightarrow \varphi^{\prime}(t) \mathrm{d} t

a. f(x, y) \rightarrow f[x(u, v), y(u, v)]
b. \iint_{D x y} \rightarrow \iint_{D u v}
c. \mathrm{d} x \mathrm{~d} y \rightarrow\left\vert\frac{\partial(x, y)}{\partial(u, v)}\right\vert \mathrm{d} u \mathrm{~d} v

\mathrm{d} x \rightarrow \varphi^{\prime}(t) \mathrm{d} t的意思是\mathrm{d} x每改变一个单位,\mathrm{d} t改变\varphi^{\prime}(t)个单位。

\mathrm{d} x \mathrm{~d} y \rightarrow\left\vert\frac{\partial(x, y)}{\partial(u, v)}\right\vert \mathrm{d} u \mathrm{~d} v的意思是\mathrm{d} x \mathrm{~d} y每改变一个单位,\mathrm{d} x \mathrm{~d} y改变\left\vert\frac{\partial(x, y)}{\partial(u,v)}\right\vert个单位。

变换前后单位向量的变换为:

(dx,0)\rightarrow (\dfrac{\partial x}{\partial u}dx,\dfrac{\partial y}{\partial u}dy)

(0,dy)\rightarrow (\dfrac{\partial x}{\partial v}dx,\dfrac{\partial y}{\partial v}dy)

变换后的面积为\left\vert\frac{\partial(x, y)}{\partial(u,v)}\right\vert

卷积公式

其实卷积公式就是二重积分换元法,例如:

Z=X+Y

可以如下换元:

\iint_{D x y} f(x, y) \mathrm{d} x \mathrm{~d} y \xlongequal[y=z-x]{x=x}\iint_{D_{x z}} f(x, z-x)\left\vert\frac{\partial(x, y)}{\partial(x, z)}\right\vert \mathrm{d} x \mathrm{~d} z

这相当于,换元后的联合概率密度g(x,z)为:

g(x,z) = f(x, z-x)\left\vert\frac{\partial(x, y)}{\partial(x, z)}\right\vert

于是我们可以用g(x,z),来求边缘概率密度f_{Z}(z)

f_{Z}(z) =\int_{-\infty}^{+\infty} g(x,z) \mathrm{d} x

(2007年真题)设二维随机变量 (X, Y) 的概率密度为

f(x, y)= \begin{cases}2-x-y, & 0<x<1,0<y<1, \\[1ex] 0, & \text { 其他. }\end{cases}

(1) 求 P\{X>2 Y\};
(2) 求 Z=X+Y 的概率密度 f_{Z}(z).

posted @ 2021-08-27 21:33:52
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