Example 3.2: Let (X, Y ) be a DRV with JPMF
fX,Y (x, y) = (
cy if x = 1, 2, . . . , n; y = 1, 2, . . . , n; x ≤ y
0 otherwise.

To solve Example 3.2, we need to analyze the **Joint Probability Mass Function (JPMF)** given for the discrete random variables \( X \) and \( Y \).

### Step 1: Understanding the JPMF

The joint probability mass function is given as:

\[
f_{X,Y}(x, y) = 
\begin{cases} 
cy & \text{if } x = 1, 2, \dots, n; y = 1, 2, \dots, n; \text{ and } x \leq y \\
0 & \text{otherwise}
\end{cases}
\]

### Step 2: Determine the Constant \( c \)

The first step in dealing with a joint PMF like this one is to determine the constant \( c \). We know that the sum of all joint probabilities must equal 1. Therefore, we sum over all valid pairs \((x, y)\) and set the result equal to 1.

\[
\sum_{x=1}^n \sum_{y=1}^n f_{X,Y}(x, y) = 1
\]

Substituting the given function \( f_{X,Y}(x, y) = cy \) where \( x \leq y \):

\[
\sum_{x=1}^n \sum_{y=x}^n cy = 1
\]

Now, the sum is taken over \( y \) starting from \( x \) to \( n \) because \( x \leq y \).

So, we can write:

\[
c \sum_{x=1}^n \sum_{y=x}^n y = 1
\]

Let's calculate the inner sum:

\[
\sum_{y=x}^n y = \frac{n(n+1)}{2} - \frac{(x-1)x}{2}
\]

This simplifies to:

\[
\sum_{y=x}^n y = \frac{n(n+1) - (x-1)x}{2} = \frac{(n-x+1)(n+x)}{2}
\]

Now the equation becomes:

\[
c \sum_{x=1}^n \frac{(n-x+1)(n+x)}{2} = 1
\]

To simplify further, calculate the sum \( \sum_{x=1}^n (n-x+1)(n+x) \) and then solve for \( c \).

### Step 3: Solve for \( c \)

Simplify the double sum, solve for \( c \), and you'll obtain the constant that normalizes the probability mass function.

### Step 4: Final JPMF

Once you have \( c \), you can express the final joint PMF \( f_{X,Y}(x, y) \).

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Would you like to go through the detailed calculation for finding the value of \( c \) or explore related questions?

### Related Questions:

1. What is the marginal PMF of \( X \) from the given joint PMF?
2. What is the marginal PMF of \( Y \) from the given joint PMF?
3. How would you calculate the conditional PMF \( f_{X|Y}(x|y) \)?
4. What is the expectation \( E[X] \) given this joint PMF?
5. How would the joint PMF change if the condition \( x \leq y \) were altered?

**Tip:** When dealing with joint distributions, always ensure the sum of probabilities equals 1.

Example 3.2: Let (X, Y) be a DRV with JPMF fX,Y (x, y) = (cy if x = 1, 2, ..., n; y = 1, 2, ..., n; x ≤ y, 0 otherwise).

This detailed solution explains how to find the constant c and the final joint probability mass function (JPMF) for two discrete random variables X and Y. The problem involves summing over valid pairs of values to ensure the probabilities sum to 1. Suitable for university-level students studying probability theory.

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