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
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:
This gives us the conditional distribution table:
| Loans Return(X₁) | −20% | 0% | 20% |
|---|---|---|---|
| P(X₁ | X₂ = 9%) | 30% | 55% |
This matches Table C, which shows the correct conditional probabilities summing to 100%.
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