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Answer: 21.0
## Explanation Since X and Y are independent, E(XY) = E(X) × E(Y). **Step 1: Calculate E(X)** - P(X=2) = 5% + 15% + 5% = 25% - P(X=7) = 10% + 30% + 10% = 50% - P(X=12) = 5% + 15% + 5% = 25% - E(X) = (2 × 0.25) + (7 × 0.50) + (12 × 0.25) = 0.5 + 3.5 + 3.0 = 7.0 **Step 2: Calculate E(Y)** - P(Y=1) = 5% + 10% + 5% = 20% - P(Y=3) = 15% + 30% + 15% = 60% - P(Y=5) = 5% + 10% + 5% = 20% - E(Y) = (1 × 0.20) + (3 × 0.60) + (5 × 0.20) = 0.2 + 1.8 + 1.0 = 3.0 **Step 3: Calculate E(XY)** - E(XY) = E(X) × E(Y) = 7.0 × 3.0 = 21.0 Therefore, the expected value of the product of X and Y is **21.0**.
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The following probability matrix contains the joint probabilities for random variables X = {2, 7, or 12} and Y = {1, 3, or 5}:
| Y | |||
|---|---|---|---|
| 1 | 3 | 5 | |
| X | |||
| 2 | 5% | 15% | 5% |
| 7 | 10% | 30% | 10% |
| 12 | 5% | 15% | 5% |
We are informed that (X) and (Y) are independent. What is the expected value of the product of X and Y, E(XY)?
A
15.0
B
21.0
C
30.5
D
35.0
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