Financial Risk Manager Part 1
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Let Y be a discrete random variable with the following probability distribution:
Y | 0 | 1 | 2 | 3
P(Y = y) | 0.3 | 0.2 | 0.4 | 0.1
Determine the variance of X, where X = 2Y + 10
Other
Community
NNikitesh
A
1.7
B
2.7
C
4
D
14
Explanation:
Step-by-step solution:
-
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
-
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)
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Calculate V[X]: V[X] = V[2Y + 10] = 4V[Y] (since V[2Y] = 4V[Y] and V[Y+10] = V[Y])
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Calculate V[Y] using formula: V[Y] = E[Y²] - (E[Y])²
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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
-
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
-
Calculate V[Y]: V[Y] = E[Y²] - (E[Y])² = 2.7 - (1.3)² = 2.7 - 1.69 = 1.01 ≈ 1
-
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
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