Math Problem Statement
Let X and Y be jointly continuous random variables with the joint density function:
f_{X,Y}(x,y) = { e^{-(x+y)}, x > 0, y > 0
0, otherwise }
(i) Compute the marginal density of Y, denoted f_Y(y). (5 points)
(ii) Compute the conditional density of X given Y, f_{X|Y}(x|y). (5 points)
Solution
Problem Recap
You are given the joint density of two jointly continuous random variables and :
\begin{cases} e^{-(x+y)}, & x > 0, \, y > 0 \\ 0, & \text{otherwise}. \end{cases}$$ You need to: 1. Compute the **marginal density of $$Y$$**, $$f_Y(y)$$. 2. Compute the **conditional density of $$X$$ given $$Y$$**, $$f_{X|Y}(x|y)$$. --- ### Part (i): Marginal Density of $$Y$$, $$f_Y(y)$$ The marginal density of $$Y$$ is obtained by integrating the joint density $$f_{X,Y}(x,y)$$ over all values of $$x$$: $$f_Y(y) = \int_{0}^{\infty} f_{X,Y}(x,y) \, dx.$$ Substitute $$f_{X,Y}(x,y) = e^{-(x+y)}$$ for $$x > 0$$ and $$y > 0$$: $$f_Y(y) = \int_{0}^{\infty} e^{-(x+y)} \, dx.$$ The exponential term can be split: $$f_Y(y) = \int_{0}^{\infty} e^{-x} e^{-y} \, dx.$$ Factor out $$e^{-y}$$ (since it does not depend on $$x$$): $$f_Y(y) = e^{-y} \int_{0}^{\infty} e^{-x} \, dx.$$ The integral of $$e^{-x}$$ over $$x \in [0, \infty)$$ is 1: $$\int_{0}^{\infty} e^{-x} \, dx = 1.$$ Thus, the marginal density of $$Y$$ is: $$f_Y(y) = e^{-y}, \quad y > 0.$$ --- ### Part (ii): Conditional Density of $$X$$ Given $$Y$$, $$f_{X|Y}(x|y)$$ The conditional density $$f_{X|Y}(x|y)$$ is defined as: $$f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}.$$ Substitute the known expressions for $$f_{X,Y}(x,y)$$ and $$f_Y(y)$$: - $$f_{X,Y}(x,y) = e^{-(x+y)}$$ for $$x > 0$$, $$y > 0$$, - $$f_Y(y) = e^{-y}$$ for $$y > 0$$. Thus: $$f_{X|Y}(x|y) = \frac{e^{-(x+y)}}{e^{-y}}.$$ Simplify the expression: $$f_{X|Y}(x|y) = e^{-x}, \quad x > 0, \, y > 0.$$ The conditional density $$f_{X|Y}(x|y)$$ is therefore: $$f_{X|Y}(x|y) = \begin{cases} e^{-x}, & x > 0, \, y > 0, \\ 0, & \text{otherwise}. \end{cases}$$ --- ### Final Answers: 1. **Marginal density of $$Y$$:** $$f_Y(y) = e^{-y}, \quad y > 0.$$ 2. **Conditional density of $$X$$ given $$Y$$:** $$f_{X|Y}(x|y) = e^{-x}, \quad x > 0, \, y > 0.$$ Let me know if you would like detailed explanations for each step! --- ### Related Questions: 1. How do you find the marginal density of $$X$$? 2. What is the expected value of $$X$$ given $$Y = y$$? 3. How do you compute the joint cumulative distribution function $$F_{X,Y}(x,y)$$? 4. What is the covariance between $$X$$ and $$Y$$? 5. How do you check if $$X$$ and $$Y$$ are independent random variables? ### Tip: To check if two random variables $$X$$ and $$Y$$ are independent, verify if their joint density can be factored into the product of their marginal densities.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Joint Density Functions
Marginal Density
Conditional Density
Formulas
Marginal Density: f_Y(y) = ∫ f_{X,Y}(x,y) dx
Conditional Density: f_{X|Y}(x|y) = f_{X,Y}(x,y) / f_Y(y)
Theorems
Joint Probability Density Properties
Law of Total Probability for Continuous Variables
Suitable Grade Level
Undergraduate Level (Probability and Statistics)
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