Financial Risk Manager Part 1
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The random variable X denotes (in units of $100,000) the size of loss per project incurred in a particular investment company. In addition, assume that X follows a chi-square distribution with 2 degrees of freedom. A risk manager randomly chooses two such projects and further assumes that their corresponding losses are independent of each other. Calculate the mean and variance of the total loss from the two projects.
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NNikitesh
A
Mean = $400,000; variance = $200,000
B
Mean = $200,000; variance = $400,000
C
Mean = $800,000; variance = $400,000
D
Mean = $400,000; variance = $800,000
Explanation:
Explanation
For a chi-square distribution with n degrees of freedom:
- Mean = n
- Variance = 2n
Given that X follows a chi-square distribution with 2 degrees of freedom:
- Mean of X = 2 (in units of
$100,000) - Variance of X = 4 (in units of
$100,000)
Since the two projects are independent:
- Mean of total loss = E(X₁ + X₂) = E(X₁) + E(X₂) = 2 + 2 = 4 (in units of
$100,000) - Variance of total loss = Var(X₁ + X₂) = Var(X₁) + Var(X₂) = 4 + 4 = 8 (in units of
$100,000)
Converting to dollars:
- Mean = 4 ×
$100,000 =$400,000 - Variance = 8 ×
$100,000 =$800,000
Alternatively, the sum of two independent chi-square variables with 2 degrees of freedom each gives a chi-square variable with 4 degrees of freedom:
- Mean = 4 (in units of
$100,000) =$400,000 - Variance = 2 × 4 = 8 (in units of
$100,000) =$800,000
Thus, option D is correct: Mean = $400,000; Variance = $800,000.
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