Find the marginal distributions of X and Y given the joint mass function  p(x, y) = \frac{C}{(x+y-1)(x+y)(x+y+1)}

Given the joint probability mass function (pmf) of \(X\) and \(Y\):

\[
p(x, y) = \frac{C}{(x+y-1)(x+y)(x+y+1)}
\]

we are tasked with finding the **marginal distributions** of \(X\) and \(Y\).

### 1. Finding the Marginal Distribution of \(X\)

To obtain the marginal distribution \(p_X(x)\), we need to sum the joint pmf \(p(x, y)\) over all possible values of \(y\):

\[
p_X(x) = \sum_y p(x, y) = \sum_y \frac{C}{(x + y - 1)(x + y)(x + y + 1)}
\]

Since the joint pmf depends on \(x + y\), let us introduce a change of variables:

Let \(z = x + y\). Then, the marginal distribution can be rewritten in terms of \(z\). For each fixed \(x\), \(y = z - x\), so we sum over \(z\):

\[
p_X(x) = \sum_{z} \frac{C}{(z - 1)z(z + 1)}
\]

This marginal distribution depends on summing over all values of \(z\) that correspond to valid values of \(y\), which are constrained by the domain of the joint distribution.

### 2. Finding the Marginal Distribution of \(Y\)

Similarly, the marginal distribution \(p_Y(y)\) is obtained by summing the joint pmf \(p(x, y)\) over all possible values of \(x\):

\[
p_Y(y) = \sum_x p(x, y) = \sum_x \frac{C}{(x + y - 1)(x + y)(x + y + 1)}
\]

As before, we let \(z = x + y\). For each fixed \(y\), \(x = z - y\), so the marginal pmf becomes:

\[
p_Y(y) = \sum_{z} \frac{C}{(z - 1)z(z + 1)}
\]

This gives a similar form to the marginal distribution of \(Y\), as it also involves summing over valid values of \(z\).

### 3. Determining the Constant \(C\)

To fully determine the marginal distributions, we must find the value of \(C\) by ensuring that the total probability sums to 1. That is, the sum of the joint distribution over all possible \(x\) and \(y\) must equal 1:

\[
\sum_x \sum_y p(x, y) = 1
\]

This step typically requires summing over the specific range of \(x\) and \(y\) based on the problem's constraints (which are not explicitly given here).

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Do you want to proceed with further steps or details on marginal distribution summation?

### Relative questions:
1. What are the valid ranges for \(x\) and \(y\) in the given joint pmf?
2. How would the constant \(C\) be determined if the ranges for \(x\) and \(y\) were known?
3. Can the sum over \(z\) be simplified for a particular range of values?
4. Are there any assumptions needed regarding the distribution type (discrete or continuous)?
5. How does the joint distribution change if constraints on \(x\) and \(y\) are introduced?

### Tip:
Always check if the joint probability mass function is normalized (i.e., sums to 1) before solving for marginal distributions.

Find the marginal distributions of X and Y given the joint mass function p(x, y) = \frac{C}{(x+y-1)(x+y)(x+y+1)}

Explore how to find the marginal distributions of X and Y from the joint mass function p(x, y) = \frac{C}{(x+y-1)(x+y)(x+y+1)}. This problem demonstrates key concepts in probability theory, such as summing over joint distributions and normalization. Ideal for undergraduate students studying probability and statistics.

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