题目3246 · 数学
Jacobian of a Scaling Map
For the transformation $x=2u,\ y=3v$, compute the absolute Jacobian determinant $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$.
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中文题目题目3247 · 数学
Jacobian of a Shear-Rotation Style Map
For $x=u+v,\ y=u-v$, compute $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$.
打开 →题目3250 · 数学
Jacobian of a Simple Nonlinear Map
For $x=u^2-v,\ y=u+v^2$, compute $\frac{\partial(x,y)}{\partial(u,v)}$ symbolically.
打开 →题目3249 · 数学
Jacobian of a Triangular Map
For $x=u,\ y=u+2v$, compute $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$.
打开 →题目3248 · 数学
Polar Coordinates Jacobian
For $x=r\cos\theta,\ y=r\sin\theta$, compute $\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right|$.
打开 →题目377 · 概率
Distribution of the Exponential of a Uniform via Jacobian
Let $X \sim \operatorname{Uniform}(0,1)$. Use the change-of-variables (Jacobian) formula to find the PDF of $Y = e^X$.
打开 →题目3265 · 数学
Area of a Polar Wedge
Use the polar Jacobian to compute the area of the region $0\le r\le 2,\ 0\le \theta\le \pi/3$.
打开 →题目3262 · 数学
Area of a Parallelogram from a Unit Square
The map $x=u+v,\ y=u-v$ sends the unit square $0\le u,v\le 1$ to a parallelogram. What is its area?
打开 →题目3261 · 数学
Area of an Ellipse from the Unit Disk
The map $x=2u,\ y=3v$ sends the unit disk $u^2+v^2\le 1$ to an ellipse in the $(x,y)$-plane. What is the area of that ellipse?
打开 →题目3264 · 数学
Area Scaling Under a Triangular Map
The map $x=u,\ y=u+2v$ sends the unit square to a parallelogram. Compute the image area.
打开 →题目3263 · 数学
Integral After a Simple Scaling
Use the change of variables $x=2u,\ y=3v$ to compute $\iint_R 1\,dx\,dy$, where $R$ is the image of the rectangle $0\le u\le 1,\ 0\le v\le 2$.
打开 →题目386 · 概率
Affine Transformation of a Normal Random Variable
Let $X \sim N(\mu, \sigma^2)$ with $a \neq 0$ and $b \in \mathbb{R}$. Using the Jacobian formula, show that $Y = aX + b$ is normally distributed and state its parameters.
打开 →题目240 · 概率
Beta Distribution and the Beta Function from Independent Gammas
Let $X \sim \text{Gamma}(\alpha, 1)$ and $Y \sim \text{Gamma}(\beta, 1)$ be independent. (a) Define $U = \frac{X}{X+Y}$ and $V = X + Y$. Compute the Jacobian of the transformation $(X, Y) \mapsto (U, V)$. (b) Derive the joint PDF of $(U, V)$ and show that $U$ and $V$ are indepe
打开 →题目385 · 概率
Box-Muller Transform: From Uniforms to Independent Normals
Let $U_1, U_2$ be independent $\operatorname{Uniform}(0,1)$ random variables. Define $$Z_1 = \sqrt{-2\ln U_1}\,\cos(2\pi U_2), \qquad Z_2 = \sqrt{-2\ln U_1}\,\sin(2\pi U_2).$$ (a) Compute the Jacobian of the inverse transformation from $(Z_1, Z_2)$ back to $(U_1, U_2)$. (b) Sho
打开 →题目400 · 概率
Deriving the Fisher F-Distribution from Chi-Squared Variables
Let $X \sim \chi^2(m)$ and $Y \sim \chi^2(n)$ be independent. Define $$F = \frac{X/m}{Y/n}.$$ (a) Using the transformation $(F, W) = \bigl(\frac{nX}{mY},\; Y\bigr)$, compute the Jacobian and derive the joint density $f_{F,W}$. (b) Integrate out $W$ to obtain the marginal PDF of
打开 →题目380 · 概率
Distribution of the Ratio of Two Independent Exponentials
Let $X$ and $Y$ be independent $\operatorname{Exp}(1)$ random variables. Define $R = X/Y$. (a) Using the transformation $(R, S) = (X/Y,\, Y)$, compute the joint density $f_{R,S}$ via the Jacobian and then marginalize over $S$ to find the PDF of $R$. (b) Identify $f_R$ as a name
打开 →题目389 · 概率
Ratio of Independent Gammas Yields a Beta Distribution
Let $X \sim \operatorname{Gamma}(\alpha, 1)$ and $Y \sim \operatorname{Gamma}(\beta, 1)$ be independent. Using the transformation $(W, S) = \bigl(X/(X+Y),\; X+Y\bigr)$: (a) Compute the Jacobian of the inverse map. (b) Derive the joint density $f_{W,S}$ and marginalize to show $
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