Financial Risk Manager Part 1

Explanation

The correct answer is D because:

1. Conditional Probability Requirements: For a valid probability distribution, all probabilities must sum to 1. In option D:

   • 0 + 0.1952 + 0.4144 + 0.3904 = 1.0000 ✓

2. Checking other options:

   • Option A: 0.0697 + 0.4274 + 0.3436 + 0.1598 = 1.0005 (slightly above 1 due to rounding)
   • Option B: 0.0697 + 0.5274 + 0.6436 + 0.1598 = 1.4005 (clearly exceeds 1)
   • Option C: 0.0697 + 0.5274 + 0.3436 + 0.2598 = 1.2005 (exceeds 1)

3. Conceptual Understanding: The conditional distribution P(X₁|X₂ = 100) represents the probability distribution of Company A's profits given that Company B made exactly 100 million in profit. This must satisfy the properties of a probability distribution:

   • Each probability must be between 0 and 1
   • The sum of all probabilities must equal 1

4. Additional Validation: Option D shows a more realistic distribution where probabilities are properly scaled and sum exactly to 1, indicating it's likely the correct conditional distribution calculated from the joint probability distribution of both companies' profits.

Key Takeaway: When evaluating conditional distributions, always verify that the probabilities sum to 1, as this is a fundamental requirement for any valid probability distribution.