The problem involves finding the probability density functions (pdfs) $f_X(x)$, $f_{Y|X}(y | 0.5)$, and $f_{X|Y}(x | 0.5)$ over specified ranges. Let's address each question and clarify the correct approach:

1. Find $f_X(x)$:

We need to define $f_X(x)$ over three intervals:

• $0 < x \leq 1$
• $1 < x < 2$
• $x < 0$ or $x \geq 2$

From the provided image:

• For $0 < x \leq 1$, $f_X(x) = \frac{x}{2}$.
• For $1 < x < 2$, the answer was incorrect. We should identify the correct function over this interval, likely based on continuity or normalization conditions.
• For $x < 0$ or $x \geq 2$, the answer provided was also incorrect, meaning it might involve setting the pdf to zero or specifying an alternative value for the non-overlapping regions.

2. Find $f_{Y|X}(y | 0.5)$:

This represents the conditional density function of $Y$ given $X = 0.5$:

• For $0 < y < 1/2$, $f_{Y|X}(y | 0.5) = 2$, which is correct.
• For $y < 0$ or $y > 1/2$, the conditional pdf might be zero outside the defined range, but 1 was marked as incorrect, indicating a different relationship or condition that needs to be considered.

3. Find $f_{X|Y}(x | 0.5)$:

This is the conditional pdf of $X$ given $Y = 0.5$:

• For $1/2 < x < 1$, $f_{X|Y}(x | 0.5) = 0.5$.
• For $1 < x < 3/2$, $f_{X|Y}(x | 0.5) = 1.5$.
• For $x < 1/2$ or $x > 3/2$, the provided answer of 0.75 was incorrect, suggesting a zero value outside this range or another specific density function.

To finalize these functions, it would be helpful to have the context of the problem or additional information like normalization conditions or the nature of the joint distribution between $X$ and $Y$.

Would you like to go through these in detail, or focus on a specific part? Here are some related questions that might help deepen understanding:

1. How do we normalize a probability density function over a given interval?
2. What are the conditions for defining a piecewise pdf?
3. How is the conditional pdf derived from joint distributions?
4. What happens to the pdf outside its defined interval?
5. How can we verify if a pdf is valid (non-negative and integrates to 1)?

Tip: When solving piecewise pdf problems, always check for continuity at the interval boundaries and ensure the entire function integrates to 1 over its defined range.