Math Problem Statement
Solution
The problem involves random variables and that are independent with an identical exponential probability distribution given by:
\begin{cases} e^{-x} & \text{if } x > 0, \\ 0 & \text{otherwise}. \end{cases}$$ We are tasked to compute the joint probability distribution of $$U = X + Y$$ and $$V = e^X$$. ### Steps to Solve: 1. **Determine the joint probability density function of $$X$$ and $$Y$$:** Since $$X$$ and $$Y$$ are independent, their joint density is: $$f_{X,Y}(x, y) = f_X(x) \cdot f_Y(y) = e^{-x} e^{-y}, \quad x, y > 0.$$ 2. **Transform the variables $$U = X + Y$$ and $$V = e^X$$:** Let: $$U = X + Y, \quad V = e^X \implies X = \ln(V), \quad Y = U - \ln(V).$$ The Jacobian of the transformation is: $$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} = \begin{vmatrix} 0 & \frac{1}{V} \\ 1 & -\frac{1}{V} \end{vmatrix} = \frac{1}{V}.$$ 3. **Substitute the densities and compute the joint PDF:** The joint density in terms of $$U$$ and $$V$$ is: $$f_{U,V}(u, v) = f_{X,Y}(x, y) \cdot \left| J \right| = f_{X,Y}(\ln(v), u - \ln(v)) \cdot \frac{1}{v}.$$ Substitute $$f_{X,Y}(x, y)$$ with the given exponential PDF: $$f_{U,V}(u, v) = e^{-\ln(v)} e^{-(u - \ln(v))} \cdot \frac{1}{v}, \quad u > \ln(v), v > 0.$$ Simplify $$e^{-\ln(v)} = \frac{1}{v}$$: $$f_{U,V}(u, v) = \frac{1}{v} \cdot e^{-u} \cdot \frac{1}{v} = \frac{e^{-u}}{v^2}, \quad u > \ln(v), v > 0.$$ Thus, the joint probability density function is: $$f_{U,V}(u, v) = \begin{cases} \frac{e^{-u}}{v^2}, & u > \ln(v), v > 0, \\ 0, & \text{otherwise}. \end{cases}$$ ### Do you need further clarifications or step-by-step verification? Here are related queries: 1. What is the marginal PDF of $$V$$? 2. How do we find the conditional PDF of $$U$$ given $$V = v$$? 3. Can you explain the physical intuition of this transformation? 4. How do the limits on $$u > \ln(v)$$ arise mathematically? 5. How is independence of $$X$$ and $$Y$$ used in this transformation? **Tip:** When transforming random variables, always compute the Jacobian and carefully identify valid ranges for the new variables.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability and Statistics
Transformation of Random Variables
Joint Probability Density Functions
Formulas
f(x) = e^(-x), x > 0
Joint PDF: f_{U,V}(u, v) = f_{X,Y}(x, y) × |Jacobian|
Jacobian determinant for transformation: |J| = 1/v
Theorems
Independence of Random Variables
Change of Variables Theorem for Joint Distributions
Suitable Grade Level
Undergraduate (Statistics/Mathematics)
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