Answer: The marginal distributions are not independent

## Explanation To determine if the marginal distributions are independent, we need to check if the joint probability distribution equals the product of the marginal distributions for all combinations of X₁ and X₂. ### Step 1: Calculate Marginal Distributions **Marginal distribution of X₁ (Company A profits):** - For X₁ = -1: 0.0197 + 0.0390 + 0.011 + 0 = 0.0697 - For X₁ = 0: 0.0395 + 0.230 + 0.127 + 0.0309 = 0.4274 - For X₁ = 2: 0.010 + 0.124 + 0.144 + 0.0656 = 0.3436 - For X₁ = 4: 0.002 + 0.0298 + 0.0662 + 0.0618 = 0.1598 **Marginal distribution of X₂ (Company B profits):** - For X₂ = -50: 0.0197 + 0.0395 + 0.010 + 0.002 = 0.0712 - For X₂ = 0: 0.0390 + 0.230 + 0.124 + 0.0298 = 0.4228 - For X₂ = 10: 0.011 + 0.127 + 0.144 + 0.0662 = 0.3482 - For X₂ = 100: 0 + 0.0309 + 0.0656 + 0.0618 = 0.1583 ### Step 2: Check Independence Condition For independence, we need: f(x₁,x₂) = f(x₁) × f(x₂) for all cells. Let's check the upper-left cell (X₁ = -1, X₂ = -50): - Joint probability: 0.0197 - Product of marginals: 0.0697 × 0.0712 = 0.00496264 Since 0.0197 ≠ 0.00496264, the distributions are not independent. ### Step 3: Verify Other Cells We could check other cells, but one counterexample is sufficient to prove lack of independence. ### Step 4: Check if Distributions are Valid All probabilities are non-negative and sum to 1: - Sum of all joint probabilities: 0.0197 + 0.0395 + 0.010 + 0.002 + 0.0390 + 0.230 + 0.124 + 0.0298 + 0.011 + 0.127 + 0.144 + 0.0662 + 0 + 0.0309 + 0.0656 + 0.0618 = 1.0000 - Each marginal distribution sums to 1 Therefore, the distributions are valid probability distributions. **Conclusion:** The marginal distributions are not independent, making option B correct.

Author: Nikitesh Somanthe