Answer-first summary for fast verification
Answer: Joint probability matrix with probabilities: (-20%, -5%) = 0.12, (-20%, 0%) = 0.093, (-20%, 9%) = 0.087, (0%, -5%) = 0.22, (0%, 0%) = 0.1705, (0%, 9%) = 0.1595, (20%, -5%) = 0.06, (20%, 0%) = 0.0465, (20%, 9%) = 0.0435
## Explanation For independent events, the joint probability is calculated by multiplying the individual probabilities: **Given probabilities:** - Loans: P(-20%) = 30% = 0.30, P(0%) = 55% = 0.55, P(20%) = 15% = 0.15 - Stock Market: P(-5%) = 40% = 0.40, P(0%) = 31% = 0.31, P(9%) = 29% = 0.29 **Joint probability calculations:** - P(-20%, -5%) = 0.30 × 0.40 = 0.12 - P(-20%, 0%) = 0.30 × 0.31 = 0.093 - P(-20%, 9%) = 0.30 × 0.29 = 0.087 - P(0%, -5%) = 0.55 × 0.40 = 0.22 - P(0%, 0%) = 0.55 × 0.31 = 0.1705 - P(0%, 9%) = 0.55 × 0.29 = 0.1595 - P(20%, -5%) = 0.15 × 0.40 = 0.06 - P(20%, 0%) = 0.15 × 0.31 = 0.0465 - P(20%, 9%) = 0.15 × 0.29 = 0.0435 Option E correctly shows these joint probabilities in decimal form, which is the standard representation for probability distributions.
Author: Tanishq Prabhu
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The resulting probability matrix displays the amount of returns of two income-generating sections of bank: Loans and Stock Market
| Loans Return | Returns(X₁) Probability | –20% | 0% | 20% |
|---|---|---|---|---|
| 30% | 55% | 15% | ||
| Stock Market Returns | Returns(X₂) Probability | –5% | 0% | 9% |
| 40% | 31% | 29% |
Assuming that the two income-generating avenues are independent of each other, what is the joint probability distribution (matrix)?
A
Joint probability matrix with probabilities: (-20%, -5%) = 12%, (-20%, 0%) = 9.3%, (-20%, 9%) = 8.7%, (0%, -5%) = 22%, (0%, 0%) = 17.05%, (0%, 9%) = 15.95%, (20%, -5%) = 6%, (20%, 0%) = 4.65%, (20%, 9%) = 4.35%
B
Joint probability matrix with probabilities: (-20%, -5%) = 30%, (-20%, 0%) = 30%, (-20%, 9%) = 30%, (0%, -5%) = 55%, (0%, 0%) = 55%, (0%, 9%) = 55%, (20%, -5%) = 15%, (20%, 0%) = 15%, (20%, 9%) = 15%
C
Joint probability matrix with probabilities: (-20%, -5%) = 12%, (-20%, 0%) = 9.3%, (-20%, 9%) = 8.7%, (0%, -5%) = 22%, (0%, 0%) = 17.05%, (0%, 9%) = 15.95%, (20%, -5%) = 6%, (20%, 0%) = 4.65%, (20%, 9%) = 4.35%
D
Joint probability matrix with probabilities: (-20%, -5%) = 70%, (-20%, 0%) = 86%, (-20%, 9%) = 44%, (0%, -5%) = 95%, (0%, 0%) = 86%, (0%, 9%) = 44%, (20%, -5%) = 70%, (20%, 0%) = 86%, (20%, 9%) = 44%
E
Joint probability matrix with probabilities: (-20%, -5%) = 0.12, (-20%, 0%) = 0.093, (-20%, 9%) = 0.087, (0%, -5%) = 0.22, (0%, 0%) = 0.1705, (0%, 9%) = 0.1595, (20%, -5%) = 0.06, (20%, 0%) = 0.0465, (20%, 9%) = 0.0435
F
Joint probability matrix with probabilities: (-20%, -5%) = 0.3, (-20%, 0%) = 0.31, (-20%, 9%) = 0.29, (0%, -5%) = 0.55, (0%, 0%) = 0.55, (0%, 9%) = 0.55, (20%, -5%) = 0.15, (20%, 0%) = 0.15, (20%, 9%) = 0.15