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The problem provided gives the joint probability mass function (PMF) table of two discrete random variables \( X \) and \( Y \), and asks which options are true based on the joint distribution and a condition \( a = \frac{b}{2} \).

### Joint PMF Table:
\[
\begin{array}{c|c|c|c}
Y \backslash X & 1 & 2 & f_X(x) \\
\hline
1 & a & \frac{1}{8} & \frac{3}{8} \\
2 & k & b & k + b \\
\hline
f_Y(y) & a+k & b + \frac{1}{8} & 1 \\
\end{array}
\]

### Key Information:
- The marginal probabilities \( f_X(x) \) and \( f_Y(y) \) are given along the rightmost column and bottom row, respectively.
- The question gives the condition \( a = \frac{b}{2} \) and asks us to determine which of the following statements are true:
  1. \( f_X(2) = f_Y(2) \)
  2. \( f_X(1) \neq f_Y(1) \)
  3. \( X \) and \( Y \) are independent.
  4. \( X \) and \( Y \) are **not** independent.

### Step 1: Marginal Probabilities
- \( f_X(1) = a + k \)
- \( f_X(2) = \frac{1}{8} + b = k + b \) (this gives us the marginal probability for \( X = 2 \)).
- \( f_Y(1) = a + k \)
- \( f_Y(2) = b + \frac{1}{8} \)

### Step 2: Independence Check
For \( X \) and \( Y \) to be independent, the joint probability must factor as the product of the marginals, i.e., \( P(X = x, Y = y) = P(X = x) \cdot P(Y = y) \). However, since the values in the table don’t consistently satisfy this property (i.e., \( a \cdot b \neq ab \) in most cases), we can conclude that **\( X \) and \( Y \) are not independent**.

### Step 3: Analyze the Statements
1. \( f_X(2) = f_Y(2) \):
   - \( f_X(2) = k + b \), and \( f_Y(2) = b + \frac{1}{8} \). 
   - These two expressions are not equal unless \( k = \frac{1}{8} \). However, with \( a = \frac{b}{2} \), it is more likely that they are **not equal**.

2. \( f_X(1) \neq f_Y(1) \):
   - \( f_X(1) = a + k \) and \( f_Y(1) = a + k \), so these **are equal**, making the statement false.

3. \( X \) and \( Y \) are independent:
   - From our analysis, \( X \) and \( Y \) are not independent, so this statement is **false**.

4. \( X \) and \( Y \) are not independent:
   - This is the correct statement. **\( X \) and \( Y \) are not independent.**

### Correct Answer:
The correct option is: **\( X \) and \( Y \) are not independent**.

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Let me know if you want further details or if you have any questions!

Here are five related questions to expand on this:
1. How is marginal probability calculated from a joint PMF table?
2. What does it mean for two random variables to be independent?
3. How do you verify independence using joint and marginal probabilities?
4. What is the significance of the condition \( a = \frac{b}{2} \) in this problem?
5. How would the results change if \( a \neq \frac{b}{2} \)?

**Tip**: Always check the sum of joint probabilities to ensure they add up to 1, as this is crucial for verifying a valid PMF!

The joint PMF table of two discrete random variables X and Y is given, and the question asks which of the following options is/are true based on a condition a = b/2.

This problem explores the joint PMF of discrete random variables X and Y, given a condition on parameters. Learn how to determine independence, calculate marginal probabilities, and analyze the relationship between X and Y.

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