https://file.aiquickdraw.com/mathbot/file/1726160508792fomi3exv.jpg

The joint probability density function (pdf) given in the image is:

\[
f(x, y) = \begin{cases} 
\frac{1}{x}, & 0 < y < x \leq 1, \\
0, & \text{otherwise}.
\end{cases}
\]

We are asked to find the marginal pdf of \(Y\), denoted as \( f_Y(y) \).

### Solution:

The marginal pdf of \(Y\) can be found by integrating the joint pdf \(f(x, y)\) with respect to \(x\) over the appropriate range:

\[
f_Y(y) = \int_{-\infty}^{\infty} f(x, y) \, dx.
\]

From the joint pdf, we know that \(f(x, y) = 0\) outside of the region where \(0 < y < x \leq 1\), so the limits for the integral will be from \(y\) to 1 (since \(x\) must be greater than \(y\) and less than or equal to 1). Therefore:

\[
f_Y(y) = \int_{y}^{1} \frac{1}{x} \, dx, \quad \text{for} \ 0 < y < 1.
\]

### Solving the integral:
\[
f_Y(y) = \int_{y}^{1} \frac{1}{x} \, dx = \left[ \ln x \right]_y^1 = \ln(1) - \ln(y) = -\ln(y).
\]

Thus, the marginal pdf of \(Y\) is:
\[
f_Y(y) = -\ln(y), \quad 0 < y < 1.
\]

### Summary:
The marginal pdf of \(Y\) is:
\[
f_Y(y) = \begin{cases} 
-\ln(y), & 0 < y < 1, \\
0, & \text{otherwise}.
\end{cases}
\]

Let me know if you'd like further details or have any questions! 

Here are some related questions you might find helpful:
1. What is the marginal pdf of \(X\)?
2. How do you find the expected value of \(Y\)?
3. Can the same process be applied to discrete random variables?
4. What is the joint cumulative distribution function (CDF) for \(X\) and \(Y\)?
5. How does the joint pdf change if the limits of \(x\) and \(y\) are altered?

**Tip:** When finding marginal distributions, always ensure you're integrating over the correct bounds based on the joint pdf's support.

Find the marginal pdf of Y given the joint pdf of random variables X and Y:
f(x, y) = { 1/x, 0 < y < x ≤ 1; 0, otherwise }.

Learn how to find the marginal pdf of Y from the given joint pdf f(x, y) = 1/x for 0 < y < x ≤ 1 using integration. This solution involves key concepts of probability and marginal distributions, with step-by-step calculations for students studying probability theory.

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