Answer: 4

**Step-by-step solution:** 1. **Given:** X = 2Y + 10, where Y has probability distribution: - P(Y=0) = 0.3 - P(Y=1) = 0.2 - P(Y=2) = 0.4 - P(Y=3) = 0.1 2. **Variance properties used:** - V[aX] = a²V[X] (where a is a constant) - V[X ± a] = V[X] (adding/subtracting a constant doesn't change variance) 3. **Calculate V[X]:** V[X] = V[2Y + 10] = 4V[Y] (since V[2Y] = 4V[Y] and V[Y+10] = V[Y]) 4. **Calculate V[Y] using formula:** V[Y] = E[Y²] - (E[Y])² 5. **Calculate E[Y]:** E[Y] = (0.3 × 0) + (0.2 × 1) + (0.4 × 2) + (0.1 × 3) = 0 + 0.2 + 0.8 + 0.3 = 1.3 6. **Calculate E[Y²]:** E[Y²] = (0.3 × 0²) + (0.2 × 1²) + (0.4 × 2²) + (0.1 × 3²) = (0.3 × 0) + (0.2 × 1) + (0.4 × 4) + (0.1 × 9) = 0 + 0.2 + 1.6 + 0.9 = 2.7 7. **Calculate V[Y]:** V[Y] = E[Y²] - (E[Y])² = 2.7 - (1.3)² = 2.7 - 1.69 = 1.01 ≈ 1 8. **Calculate V[X]:** V[X] = 4V[Y] = 4 × 1 = 4 **Key points:** - Variance is unaffected by adding/subtracting constants - Multiplying by a constant scales variance by the square of that constant - The variance calculation shows how linear transformations affect dispersion

Author: Nikitesh Somanthe