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The problem in the image asks us to find the marginal probability density function (p.d.f.) \( f_X(x) \) of the random variable \( X \), given the joint probability density function \( f(x, y) = 6y \) for \( 0 \leq x \leq y \leq 1 \).

### To solve this:
1. The marginal p.d.f. of \( X \), denoted by \( f_X(x) \), can be found by integrating the joint p.d.f. \( f(x, y) \) over all possible values of \( y \) for a given \( x \).
   
   This is done using the formula:
   \[
   f_X(x) = \int_{y_{\text{min}}}^{y_{\text{max}}} f(x, y) \, dy
   \]

2. Here, the limits of integration depend on the condition \( 0 \leq x \leq y \leq 1 \). Therefore, for a fixed \( x \), \( y \) ranges from \( x \) to 1.

3. Substituting the given joint p.d.f. \( f(x, y) = 6y \), we get:
   \[
   f_X(x) = \int_{x}^{1} 6y \, dy
   \]

4. Now, perform the integration:
   \[
   f_X(x) = 6 \int_{x}^{1} y \, dy = 6 \left[ \frac{y^2}{2} \right]_{x}^{1}
   \]
   \[
   f_X(x) = 6 \left( \frac{1^2}{2} - \frac{x^2}{2} \right) = 6 \cdot \frac{1 - x^2}{2} = 3(1 - x^2)
   \]

Thus, the marginal p.d.f. of \( X \) is:
\[
f_X(x) = 3(1 - x^2), \quad 0 \leq x \leq 1
\]

Would you like further clarification on any step?

Here are some related questions to consider:
1. How do you calculate the joint probability density function from marginal p.d.f.s?
2. What are the properties of a marginal p.d.f.?
3. How does the support of the joint p.d.f. affect the limits of integration?
4. What are other methods for finding marginal distributions besides integration?
5. How is the concept of a conditional distribution related to the marginal p.d.f.?

**Tip:** When solving for marginal p.d.f.s, always check the bounds of integration carefully, as they depend on the support of the random variables involved.

Given f(x, y) = 6y for 0 ≤ x ≤ y ≤ 1, find the marginal p.d.f. f_X(x) of X.

Learn how to calculate the marginal probability density function f_X(x) for X, given the joint p.d.f. f(x, y) = 6y for 0 ≤ x ≤ y ≤ 1. This problem involves key concepts in probability theory such as marginal and joint distributions, suitable for university-level students.

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