Answer: 24.56

## Explanation To calculate the covariance between company A and B, we need to use the formula: **Cov(X₁, X₂) = E(X₁X₂) - E(X₁)E(X₂)** We are given that **E(X₁X₂) = 43.23**, so we need to calculate **E(X₁)** and **E(X₂)** from the joint probability distribution table. ### Step 1: Calculate E(X₁) - Expected value of Company A profits First, we need to find the marginal distribution of X₁ by summing probabilities across each column (for each value of X₁). **Values of X₁:** -1, 0, 2, 4 (in millions) **Marginal probabilities for X₁:** - 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 **Check sum:** 0.0697 + 0.4274 + 0.3436 + 0.1598 = 1.0005 (≈ 1, minor rounding differences) **E(X₁) = Σ [x₁ × P(X₁ = x₁)]** = (-1 × 0.0697) + (0 × 0.4274) + (2 × 0.3436) + (4 × 0.1598) = -0.0697 + 0 + 0.6872 + 0.6392 = **1.2567** ### Step 2: Calculate E(X₂) - Expected value of Company B profits Now find the marginal distribution of X₂ by summing probabilities across each row (for each value of X₂). **Values of X₂:** -50, 0, 10, 100 (in millions) **Marginal probabilities for X₂:** - 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 **Check sum:** 0.0712 + 0.4228 + 0.3482 + 0.1583 = 1.0005 (≈ 1) **E(X₂) = Σ [x₂ × P(X₂ = x₂)]** = (-50 × 0.0712) + (0 × 0.4228) + (10 × 0.3482) + (100 × 0.1583) = -3.56 + 0 + 3.482 + 15.83 = **15.752** ### Step 3: Calculate Covariance **Cov(X₁, X₂) = E(X₁X₂) - E(X₁)E(X₂)** = 43.23 - (1.2567 × 15.752) = 43.23 - 19.796 = **23.434** Rounding to two decimal places: **23.43** Therefore, the correct answer is **B. 23.43**. **Note:** The question states that E(X₁X₂) = 43.23 is given, which saves us from having to calculate it directly from the joint distribution table. This value would be computed as ΣΣ x₁x₂ × P(X₁=x₁, X₂=x₂) across all combinations in the table.

Author: Nikitesh Somanthe