Answer-first summary for fast verification
Answer: Table C
## Explanation The correct answer is **C** because the conditional distribution of X₁ given X₂ = 9% is calculated by dividing the joint probabilities in the row where X₂ = 9% by the marginal probability of X₂ = 9% (which is 29%). **Calculation:** - For X₁ = -20%: 8.7% ÷ 29% = 0.3 = 30% - For X₁ = 0%: 15.95% ÷ 29% = 0.55 = 55% - For X₁ = 20%: 4.35% ÷ 29% = 0.15 = 15% This gives us the conditional distribution table: | Loans Return(X₁) | −20% | 0% | 20% | |------------------|------|-----|-----| | P(X₁|X₂ = 9%) | 30% | 55% | 15% | This matches Table C, which shows the correct conditional probabilities summing to 100%.
Author: Tanishq Prabhu
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The conditional distribution of X₁ given X₂ is defined as:
So, in this case, we need:
Note that we are given the marginal distribution of stock market return f_{(X₂)}(x₂) given by:
| Stock Market Returns | Returns(X₂) | −5% | 0% | 9% |
|---|---|---|---|---|
| Probability | 40% | 31% | 29% |
We calculated the joint distribution as:
| Loan Return (X₁) | |||
|---|---|---|---|
| Stock Market Returns(X₂) | −20% | 0% | 20% |
| −5% | 12% | 22% | 6% |
| 0% | 9.3% | 17.05% | 4.65% |
| 9% | 8.7% | 15.95% | 4.35% |
To calculate the conditional distribution, we divide the last column by the corresponding marginal distribution (29%). So, the conditional distribution is:
A
Table A
B
Table B
C
Table C
D
Table D