https://r2.erweima.ai/mathbot/image/17227393046271g9a2m7c.jpg

find marginal density of X. 
ans is 4x^3
give step by step answer

To find the marginal density of \( X \), we need to integrate the joint probability density function \( f_{X,Y}(x,y) \) over all possible values of \( Y \). The joint probability density function \( f_{X,Y}(x,y) \) is given by:

\[
f_{X,Y}(x,y) = \begin{cases} 
8xy & \text{if } 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1, \\ 
0 & \text{otherwise}.
\end{cases}
\]

The marginal density function of \( X \), denoted by \( f_X(x) \), is obtained by integrating \( f_{X,Y}(x,y) \) with respect to \( y \):

\[
f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) \, dy.
\]

Given the support of \( f_{X,Y}(x,y) \), we only need to integrate from \( 0 \) to \( 1 \):

\[
f_X(x) = \int_{0}^{1} 8xy \, dy.
\]

Let's perform the integration step by step:

1. Set up the integral:

\[
f_X(x) = \int_{0}^{1} 8xy \, dy.
\]

2. Integrate \( 8xy \) with respect to \( y \):

\[
f_X(x) = 8x \int_{0}^{1} y \, dy.
\]

3. Compute the integral \( \int_{0}^{1} y \, dy \):

\[
\int_{0}^{1} y \, dy = \left[ \frac{y^2}{2} \right]_{0}^{1} = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}.
\]

4. Multiply the result by \( 8x \):

\[
f_X(x) = 8x \cdot \frac{1}{2} = 4x.
\]

Therefore, the marginal density function of \( X \) is:

\[
f_X(x) = 4x \quad \text{for } 0 \leq x \leq 1.
\]

If you need any more details or have any questions, feel free to ask.

Here are some related questions for further exploration:

1. What is the marginal density of \( Y \)?
2. How do you find the expected value of \( X \) given the marginal density?
3. How do you find the variance of \( X \) given the marginal density?
4. How do you find the covariance between \( X \) and \( Y \)?
5. What is the joint cumulative distribution function (CDF) of \( X \) and \( Y \)?
6. How do you find the conditional density \( f_{Y|X}(y|x) \)?
7. What is the correlation coefficient between \( X \) and \( Y \)?
8. How do you determine if \( X \) and \( Y \) are independent?

**Tip:** Understanding how to derive marginal and conditional densities from a joint density function is crucial for analyzing the relationships between multiple random variables.

Learn how to find the marginal density of X from a joint probability density function with a detailed step-by-step solution. This problem involves integrating the joint density function over the support of X. Suitable for advanced undergraduate students and anyone studying probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation