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To solve for the covariance \( \text{cov}(X, Y) \) of the discrete random variables \( X \) and \( Y \), we need to follow these steps:

1. **Covariance Formula**: 
   \[
   \text{cov}(X, Y) = E[XY] - E[X]E[Y]
   \]
   where \( E[XY] \) is the expected value of the product of \( X \) and \( Y \), and \( E[X] \) and \( E[Y] \) are the expected values of \( X \) and \( Y \), respectively.

2. **Expected Value of \( X \) and \( Y \)**:
   \[
   E[X] = \sum_{x} x \cdot P(X = x), \quad E[Y] = \sum_{y} y \cdot P(Y = y)
   \]
   To find these, we need to sum over the marginal probabilities of \( X \) and \( Y \).

3. **Expected Value of the Product \( E[XY] \)**:
   \[
   E[XY] = \sum_{x} \sum_{y} x \cdot y \cdot P(X = x, Y = y)
   \]

Let's compute this step by step using the given joint probability table:

### Step 1: Marginal Probabilities
#### Marginal Probability \( P(X = x) \)
- \( P(X = 0) = 0.1 + 0.2 = 0.3 \)
- \( P(X = 1) = 0.2 + 0.1 = 0.3 \)
- \( P(X = 2) = 0.3 + 0.1 = 0.4 \)

#### Marginal Probability \( P(Y = y) \)
- \( P(Y = -1) = 0.1 + 0.2 + 0.3 = 0.6 \)
- \( P(Y = 2) = 0.2 + 0.1 + 0.1 = 0.4 \)

### Step 2: Expected Values
#### \( E[X] \)
\[
E[X] = 0 \cdot 0.3 + 1 \cdot 0.3 + 2 \cdot 0.4 = 0.0 + 0.3 + 0.8 = 1.1
\]

#### \( E[Y] \)
\[
E[Y] = (-1) \cdot 0.6 + 2 \cdot 0.4 = -0.6 + 0.8 = 0.2
\]

### Step 3: Compute \( E[XY] \)
We multiply each \( X \) and \( Y \) value by their joint probabilities:
\[
E[XY] = (0 \cdot (-1) \cdot 0.1) + (0 \cdot 2 \cdot 0.2) + (1 \cdot (-1) \cdot 0.2) + (1 \cdot 2 \cdot 0.1) + (2 \cdot (-1) \cdot 0.3) + (2 \cdot 2 \cdot 0.1)
\]
\[
E[XY] = 0 + 0 + (-0.2) + 0.2 + (-0.6) + 0.4 = -0.2
\]

### Step 4: Compute \( \text{cov}(X, Y) \)
\[
\text{cov}(X, Y) = E[XY] - E[X]E[Y] = -0.2 - (1.1)(0.2) = -0.2 - 0.22 = -0.42
\]

Thus, the covariance is \( \text{cov}(X, Y) = -0.42 \), corresponding to option **a**.

Would you like further details or clarification on any of the steps?

### Related Questions:
1. How do you calculate the variance of \( X \) and \( Y \) from the joint probability distribution?
2. What does the sign of the covariance tell us about the relationship between \( X \) and \( Y \)?
3. How can you use the covariance to find the correlation coefficient?
4. What is the difference between marginal probability and conditional probability?
5. How would the covariance change if the joint probability distribution were different?

### Tip:
Always check that the joint probabilities sum to 1 when dealing with probability distributions to ensure consistency.

Calculate the covariance \(\text{cov}(X, Y)\) given the joint probability density function of discrete random variables X and Y, represented in a table.

Learn how to calculate the covariance of two discrete random variables X and Y using their joint probability density function. This problem involves calculating expected values and applying the covariance formula, suitable for undergraduate-level probability and statistics.

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