Explanation

Since X and Y are independent, we can use the property that for independent random variables:

$E(XY) = E(X) \cdot E(Y)$

Step 1: Calculate E(X)

Marginal probabilities for X:

• P(X=2) = 5% + 15% + 5% = 25%
• P(X=7) = 10% + 30% + 10% = 50%
• P(X=12) = 5% + 15% + 5% = 25%

$E(X) = 2 \times 0.25 + 7 \times 0.50 + 12 \times 0.25 = 0.5 + 3.5 + 3.0 = 7.0$

Step 2: Calculate E(Y)

Marginal probabilities for Y:

• P(Y=1) = 5% + 10% + 5% = 20%
• P(Y=3) = 15% + 30% + 15% = 60%
• P(Y=5) = 5% + 10% + 5% = 20%

$E(Y) = 1 \times 0.20 + 3 \times 0.60 + 5 \times 0.20 = 0.2 + 1.8 + 1.0 = 3.0$

Step 3: Calculate E(XY)

$E(XY) = E(X) \cdot E(Y) = 7.0 \times 3.0 = 21.0$

However, let me verify this by direct calculation:

$E(XY) = \sum_{x,y} x \cdot y \cdot P(X=x, Y=y)$

$E(XY) = (2\times1\times0.05) + (2\times3\times0.15) + (2\times5\times0.05) + (7\times1\times0.10) + (7\times3\times0.30) + (7\times5\times0.10) + (12\times1\times0.05) + (12\times3\times0.15) + (12\times5\times0.05)$

$E(XY) = 0.1 + 0.9 + 0.5 + 0.7 + 6.3 + 3.5 + 0.6 + 5.4 + 3.0 = 21.0$

Wait, I made an error in my initial calculation. The direct calculation gives 21.0, which matches option B.

Correction: The correct answer is B. 21.0

For independent random variables, E(XY) = E(X) × E(Y) = 7.0 × 3.0 = 21.0, which is confirmed by the direct calculation.