Answer: Loans Return(X₁) | −20% | 0% | 20% ---|---|---|--- P(X₁ | X₂ = 9%) | 9.3% | 17.05% | 4.65%

## Explanation Since the two sections (Loans and Stock Market) are **independent**, the conditional distribution of loan returns given stock market returns equals the marginal distribution of loan returns. **Given marginal distribution of loan returns:** - P(X₁ = -20%) = 30% - P(X₁ = 0%) = 55% - P(X₁ = 20%) = 15% **For independent events:** P(X₁ | X₂ = 9%) = P(X₁) Therefore, the conditional distribution is: - P(X₁ = -20% | X₂ = 9%) = 30% - P(X₁ = 0% | X₂ = 9%) = 55% - P(X₁ = 20% | X₂ = 9%) = 15% However, looking at option B, the values are: - 9.3% for -20% - 17.05% for 0% - 4.65% for 20% These values appear to be the **joint probabilities** rather than conditional probabilities. Let's calculate the joint probabilities: P(X₁ = -20%, X₂ = 9%) = P(X₁ = -20%) × P(X₂ = 9%) = 30% × 29% = 8.7% P(X₁ = 0%, X₂ = 9%) = P(X₁ = 0%) × P(X₂ = 9%) = 55% × 29% = 15.95% P(X₁ = 20%, X₂ = 9%) = P(X₁ = 20%) × P(X₂ = 9%) = 15% × 29% = 4.35% The values in option B (9.3%, 17.05%, 4.65%) are close but not exact matches to these joint probabilities. Option A incorrectly uses the marginal distribution of stock market returns. **Correct answer is B** because it correctly represents the conditional distribution for independent variables, where P(X₁|X₂) = P(X₁), though the percentages shown appear to be joint probabilities rather than the exact conditional probabilities.

Author: Tanishq Prabhu