University Statistics Test Questions on Bivariate Jacobian Transformations

📚 Quick Study Guide

• 📏 Definition: The Bivariate Jacobian Transformation is used to find the joint probability density function (PDF) of two transformed random variables.
• ➗ Formula Overview: If $U = g(X, Y)$ and $V = h(X, Y)$, then the joint PDF of $U$ and $V$, denoted as $f_{U,V}(u, v)$, is given by: $f_{U,V}(u, v) = f_{X,Y}(x(u, v), y(u, v)) |J|$, where $J$ is the Jacobian determinant.
• 🧮 Jacobian Determinant: The Jacobian determinant, $J$, is calculated as: $J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}$
• 📝 Steps:
  1. Express $x$ and $y$ in terms of $u$ and $v$: $x = x(u, v)$ and $y = y(u, v)$.
  2. Compute the partial derivatives needed for the Jacobian determinant.
  3. Calculate the Jacobian determinant $J$.
  4. Find the absolute value of $J$, i.e., $|J|$.
  5. Substitute $x(u, v)$, $y(u, v)$, and $|J|$ into the original joint PDF $f_{X,Y}(x, y)$.
  6. Determine the new support/region for $U$ and $V$.
• 💡 Important Note: The transformation must be one-to-one (invertible) within the region of interest.

🧪 Practice Quiz

1. Question 1: If $X$ and $Y$ are random variables and $U = X + Y$, $V = X - Y$, what is the Jacobian determinant $J$ for the transformation from $(X, Y)$ to $(U, V)$?
   1. A) 0.5
   2. B) -0.5
   3. C) 1
   4. D) -1
2. Question 2: Given $X$ and $Y$ are independent random variables with joint PDF $f_{X,Y}(x, y) = e^{-(x+y)}$ for $x > 0, y > 0$, and the transformation $U = X + Y$, $V = X / Y$, what is a key step in finding the PDF of $U$ and $V$?
   1. A) Ignoring the Jacobian determinant.
   2. B) Computing the Jacobian determinant.
   3. C) Setting $V = 0$.
   4. D) Assuming $U$ and $V$ are independent.
3. Question 3: If $U = X^2$ and $V = Y^2$, and you need to find the Jacobian determinant, what partial derivatives are required?
   1. A) $\frac{\partial X}{\partial U}, \frac{\partial Y}{\partial V}$
   2. B) $\frac{\partial U}{\partial X}, \frac{\partial V}{\partial Y}$
   3. C) $\frac{\partial X}{\partial U}, \frac{\partial X}{\partial V}, \frac{\partial Y}{\partial U}, \frac{\partial Y}{\partial V}$
   4. D) $\frac{\partial U}{\partial X}, \frac{\partial U}{\partial Y}, \frac{\partial V}{\partial X}, \frac{\partial V}{\partial Y}$
4. Question 4: Let $X$ and $Y$ be jointly distributed with some PDF. If you apply a bivariate transformation and the Jacobian determinant $J = 0$, what does this imply?
   1. A) The transformation is valid and the PDF can be found easily.
   2. B) The transformation is not one-to-one, and the method is not applicable.
   3. C) The new variables are independent.
   4. D) The original variables were independent.
5. Question 5: Consider $X$ and $Y$ with joint PDF $f(x, y)$. If $U = g(X, Y)$ and $V = h(X, Y)$, what do you substitute into the original joint PDF after calculating the Jacobian?
   1. A) $u$ and $v$ directly.
   2. B) $g(u, v)$ and $h(u, v)$.
   3. C) $x(u, v)$ and $y(u, v)$.
   4. D) Constants.
6. Question 6: The Jacobian transformation method is MOST useful when you need to find:
   1. A) The marginal PDF of a single variable directly.
   2. B) The joint PDF of transformed random variables.
   3. C) The expected value without transforming variables.
   4. D) The correlation between variables without transformation.
7. Question 7: Suppose $U = XY$ and $V = X/Y$. What is a common first step after finding the Jacobian?
   1. A) Integrate over all possible values of X and Y.
   2. B) Express X and Y in terms of U and V.
   3. C) Differentiate U and V with respect to X and Y.
   4. D) Set U and V to zero.