This question asks about the conditional probability density function $f_{X|Y}(x|y)$. The correct formula for $f_{X|Y}(x|y)$ is based on the joint and marginal distributions.

Key Concept:

The conditional probability density function is defined as: $f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)},$ where:

• $f_{X,Y}(x,y)$ is the joint probability density function of $X$ and $Y$,
• $f_Y(y)$ is the marginal density function of $Y$.

Analysis of Choices:

• Option a: "None of the above" is incorrect because one of the options is correct.
• Option b: $f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$ is correct as this is the standard definition of conditional probability density.
• Option c: $f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_X(x)}$ is incorrect. The denominator should be $f_Y(y)$, not $f_X(x)$.
• Option d: $f_{X|Y}(x|y) = \frac{f_X(x)}{f_Y(y)}$ is incorrect. This equation does not align with the definition of conditional density.

Final Answer:

The correct answer is b.

Let me know if you'd like further clarification or if you have additional questions. Here are some related questions to expand on this topic:

1. How do you compute marginal density from a joint probability distribution?
2. What is the relationship between independence and conditional density?
3. How does $f_{X|Y}(x|y)$ differ when $X$ and $Y$ are independent?
4. Can conditional density $f_{X|Y}(x|y)$ ever be greater than the joint density $f_{X,Y}(x,y)$?
5. Why is the marginal density $f_Y(y)$ used as the denominator?

Tip: Always verify the validity of a formula by considering the integral properties of probability densities.