Financial Risk Manager Part 1
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The yearly profits of the two firms A and B can be summarized in the following probability matrix.
| Company A (X₁) Profits | ||||
|---|---|---|---|---|
| Company B (X₂) Profits | -1 Million | 0 Million | 2 Million | 4 Million |
| -50 Million | 0.0197 | 0.0395 | 0.010 | 0.002 |
| 0 Million | 0.0390 | 0.230 | 0.124 | 0.0298 |
| 10 Million | 0.011 | 0.127 | 0.144 | 0.0662 |
| 100 Million | 0 | 0.0309 | 0.0656 | 0.0618 |
Looking at the marginal distribution of companies A and B, which of the following of the statements is true?
Other
Community
NNikitesh
A
The marginal distributions are independent
B
The marginal distributions are not independent
C
The distributions are not ideal probability distributions
D
None of the above
Explanation:
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.
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