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To find the marginal density of \( X \) from the given joint probability density function (pdf) \( f_{X,Y}(x,y) \), we integrate the joint pdf over the range of the variable \( Y \).

Given the joint pdf:
\[ f_{X,Y}(x,y) = \begin{cases} 
8xy, & 0 \leq x \leq 1, 0 \leq y \leq 1 \\
0, & \text{otherwise} 
\end{cases} \]

The marginal density of \( X \), denoted as \( f_X(x) \), is calculated as follows:
\[ f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) \, dy \]

Since the joint pdf is non-zero only in the range \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \), we can adjust the limits of integration accordingly:
\[ f_X(x) = \int_{0}^{1} 8xy \, dy \]

Now, we calculate the integral:
\[
f_X(x) = \int_{0}^{1} 8xy \, dy
= 8x \int_{0}^{1} y \, dy
= 8x \left[\frac{y^2}{2}\right]_0^1
= 8x \left(\frac{1}{2} - 0\right)
= 4x
\]

Thus, the marginal density of \( X \) is:
\[ f_X(x) = \begin{cases} 
4x, & 0 \leq x \leq 1 \\
0, & \text{otherwise} 
\end{cases} \]

Would you like further details or have any other questions?

**Related Questions:**
1. What is the marginal density of \( Y \)?
2. How do you find the expected value of a continuous random variable?
3. What is the joint density function of two independent continuous random variables?
4. How is the joint pdf normalized to ensure it integrates to 1 over all possible values?
5. What are the conditions for a function to be a valid joint pdf?
6. How do you interpret the marginal density in the context of probability?
7. What is the role of the limits in the integration for finding marginal densities?
8. How do you determine the range for the integration when finding marginal densities?

**Tip:** When working with joint density functions, always ensure the function is properly normalized by integrating over the entire space and verifying that the result equals 1.

Learn how to find the marginal density of X from a given joint probability density function (pdf). This problem involves integrating the joint pdf over the range of the variable Y to obtain the marginal density of X. Suitable for advanced undergraduate students and anyone interested in probability theory and density functions.

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