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A distribution x₁, x₂, x₃,..., xₙ has variance 10. What is the variance of the new distribution 3x₁ + 2, 3x₂ + 2, 3x₃ + 2,..., 3xₙ + 2?

1. 10
2. 30
3. 10/3
4. 90

Correct answer: 90

Solution

The transformation y_i = 3x_i + 2 is a linear shift and scale. The constant +2 does not affect variance, while multiplying by 3 scales variance by 3² = 9. So the new variance = 9 * 10 = 90.

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