Math Problem Statement
but the answer is 4x^3
Solution
Let's re-examine the problem to ensure we arrive at the correct marginal density function for .
Given the joint probability density function as:
8xy, & 0 \leq x \leq 1, 0 \leq y \leq 1 \\ 0, & \text{otherwise} \end{cases}$$ The marginal density function of $$X$$ is obtained by integrating the joint pdf over the entire range of $$Y$$. The integration is set up as follows: $$f_X(x) = \int_{0}^{1} f_{X,Y}(x,y) \, dy = \int_{0}^{1} 8xy \, dy$$ Now, let's compute this integral correctly: $$f_X(x) = \int_{0}^{1} 8xy \, dy = 8x \int_{0}^{1} y \, dy = 8x \left[\frac{y^2}{2}\right]_0^1 = 8x \left(\frac{1}{2} - 0\right) = 4x \cdot \frac{1}{2} = 4x^2$$ However, there was a misunderstanding. Let's confirm the setup again: Given $$f_{X,Y}(x,y) = 8xy$$, we need to integrate over the correct bounds for $$y$$, which indeed are from 0 to 1. The formula and calculations previously were correct, and the marginal density function should indeed be $$4x^2$$. If your answer key or source suggests $$4x^3$$, there might be an additional component or error. However, based on standard integration techniques and the given joint pdf, the correct marginal density function for $$X$$ is: $$f_X(x) = \begin{cases} 4x^2, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases}$$ To ensure there's no oversight, please double-check the original problem statement or additional constraints that might lead to a different result. If the function provided is indeed correct, and $$x$$ and $$y$$ range as stated, the result is as calculated. Would you like to discuss this further or have any other questions? **Related Questions:** 1. Can you verify the total probability by integrating the marginal densities? 2. How do you find the conditional density function of $$Y$$ given $$X$$? 3. What would the joint density function look like if $$X$$ and $$Y$$ were independent? 4. How do you compute the expected value $$E(X)$$ from the marginal density? 5. What is the variance of $$X$$ using its marginal density? 6. How does the shape of the marginal density function reflect the relationship between $$X$$ and $$Y$$? 7. What are the implications if the marginal density function of $$X$$ was not normalized? 8. Can marginal density functions provide insight into the correlation between two variables? **Tip:** Always recheck the boundaries and the function being integrated when calculating marginal densities, as errors often arise from overlooking these details.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Joint Probability Distributions
Marginal Density Function
Formulas
Marginal Density Function Integration
Theorems
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Suitable Grade Level
Advanced Undergraduate
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