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.