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The problem states that the joint probability mass function (p.m.f.) of \( X \) and \( Y \) is given by:

\[
p(x, y) = k(2x + 3y), \quad x = 0, 1, 2; \, y = 1, 2, 3.
\]

We are tasked with:

1. Finding the **marginal distributions** of \( X \) and \( Y \).
2. Finding the **conditional probability distributions**.
3. Finding the **probability distribution of \( X + Y \)**.

Let's go step by step:

### 1. Finding the constant \( k \):
Since the total probability must sum to 1, we can use the following equation:

\[
\sum_{x=0}^{2} \sum_{y=1}^{3} p(x, y) = 1.
\]

Substitute \( p(x, y) = k(2x + 3y) \) and sum over all possible values of \( x \) and \( y \):

\[
\sum_{x=0}^{2} \sum_{y=1}^{3} k(2x + 3y) = 1.
\]

Now, I'll calculate this to find the value of \( k \).The constant \( k \) is \( \frac{1}{72} \).

Now that we have \( k = \frac{1}{72} \), let's move to the next steps.

### 2. Marginal Distributions

The **marginal distribution of \( X \)** is found by summing the joint probability over all values of \( y \):

\[
p_X(x) = \sum_{y=1}^{3} p(x, y).
\]

Similarly, the **marginal distribution of \( Y \)** is found by summing the joint probability over all values of \( x \):

\[
p_Y(y) = \sum_{x=0}^{2} p(x, y).
\]

I'll calculate the marginal distributions for both \( X \) and \( Y \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually calculate the marginal distributions using the formula:

1. **Marginal Distribution of \( X \)**:
   \[
   p_X(x) = \sum_{y=1}^{3} p(x, y)
   \]
   where \( p(x, y) = \frac{1}{72} \cdot (2x + 3y) \).

2. **Marginal Distribution of \( Y \)**:
   \[
   p_Y(y) = \sum_{x=0}^{2} p(x, y)
   \]

Let me know if you'd like to continue solving it step by step!

### Here are some related questions to explore:
1. How are marginal distributions used in probability theory?
2. What are conditional probability distributions?
3. How can you derive the probability distribution of a sum of two random variables?
4. What are joint probability mass functions used for in real-world applications?
5. How does knowing \( p(x, y) \) help in calculating expectations?

#### Tip:
Always check that the sum of probabilities equals 1 to ensure your probability distribution is valid.

The joint p.m.f. of (X, Y) is given by p(x, y) = k(2x + 3y), x = 0, 1, 2; y = 1, 2, 3. Find all the marginal, conditional probability distributions. Find the probability distribution of (X + Y).

Explore how to calculate the marginal and conditional distributions of discrete random variables X and Y, given their joint probability mass function p(x, y) = k(2x + 3y). Learn how to find the probability distribution of X + Y. Suitable for students in undergraduate probability courses.

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