Financial Risk Manager Part 1
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Two random variables X and Y are such that V[X] = 4V[Y] and Cov[X,Y] = V[Y]. Let E = X + Y and F = X - Y. Find Cov[E, F].
Other
Community
NNikitesh
A
V[Y] - V[X]
B
Cov[X,Y]
C
V[Y]
D
3V[Y]
Explanation:
Step-by-step solution:
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Given relationships:
- V[X] = 4V[Y]
- Cov[X,Y] = V[Y]
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Define E and F:
- E = X + Y
- F = X - Y
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Calculate Cov[E, F]: Cov[E, F] = Cov[X + Y, X - Y]
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Expand using covariance properties: Cov[X + Y, X - Y] = Cov[X, X] - Cov[X, Y] + Cov[Y, X] - Cov[Y, Y]
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Simplify using covariance properties:
- Cov[X, X] = V[X]
- Cov[Y, X] = Cov[X, Y] (covariance is symmetric)
- Cov[Y, Y] = V[Y]
So: Cov[E, F] = V[X] - Cov[X,Y] + Cov[X,Y] - V[Y]
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Cancel terms: Cov[E, F] = V[X] - V[Y]
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Substitute given relationship V[X] = 4V[Y]: Cov[E, F] = 4V[Y] - V[Y] = 3V[Y]
Key concepts used:
- Covariance properties: Cov(aX + bY, cW + dZ) = acCov(X,W) + adCov(X,Z) + bcCov(Y,W) + bdCov(Y,Z)
- Cov(X,X) = V[X]
- Cov(X,Y) = Cov(Y,X) (symmetry)
- Variance is a special case of covariance where both variables are the same
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