To solve this problem, let's establish a general understanding of conditional probability density functions for random variables $X$ and $Y$.

Given that we want:

1. The conditional probability density function of $X$ when $Y = 1$: $f_{X|Y}(x|Y=1)$.
2. The conditional probability density function of $Y$ when $X = 2$: $f_{Y|X}(y|X=2)$.

These conditional probability density functions rely on the marginal probabilities of $X$ and $Y$, which we denote as $p(x_i)$ for $X$ and $q(y_j)$ for $Y$. These marginals are essentially the overall probabilities for each variable independently.

Steps for Solution

1. Marginal Probabilities $p(x_i)$ and $q(y_j)$:

   • $p(x_i) = P(X = x_i)$: This is obtained by summing or integrating over all values of $Y$.
   • $q(y_j) = P(Y = y_j)$: This is obtained by summing or integrating over all values of $X$.

2. Conditional Probability Density Function $f_{X|Y}(x|Y=1)$:

   • This is calculated as: $f_{X|Y}(x | Y=1) = \frac{f_{X,Y}(x, Y=1)}{q(y=1)}$ where $f_{X,Y}(x, Y=1)$ is the joint probability density of $X$ and $Y$ evaluated at $Y=1$, and $q(y=1)$ is the marginal probability of $Y=1$.

3. Conditional Probability Density Function $f_{Y|X}(y|X=2)$:

   • This is calculated as: $f_{Y|X}(y | X=2) = \frac{f_{X,Y}(X=2, y)}{p(x=2)}$ where $f_{X,Y}(X=2, y)$ is the joint probability density of $X$ and $Y$ evaluated at $X=2$, and $p(x=2)$ is the marginal probability of $X=2$.

By determining the marginal probabilities $p(x_i)$ and $q(y_j)$, we can compute these conditional densities based on the given values or the joint distribution of $X$ and $Y$.

Would you like more detailed steps or any specific values computed here?

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Here are five additional related questions:

1. How are joint probability distributions used in conditional probability?
2. What if $X$ and $Y$ are independent; how does that affect the conditional densities?
3. How can we use Bayes' Theorem with these conditional probability densities?
4. How are marginal probabilities obtained if the probability distribution is discrete versus continuous?
5. What is the significance of conditional density functions in real-world applications?

Tip: When dealing with conditional probabilities, always identify marginal distributions, as they are essential in deriving the conditional probabilities.