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