Financial Risk Manager Part 1
Get started today
Ultimate access to all questions.
The following are the two series of data X and Y.
| X | Y |
|---|---|
| 700 | 318 |
| 130 | 304 |
| 140 | 317 |
| 200 | 305 |
| 230 | 309 |
| 250 | 307 |
| 270 | 316 |
| 340 | 309 |
| 400 | 315 |
| 620 | 327 |
| 620 | 450 |
| 760 | 324 |
| 650 | 500 |
| 750 | 699 |
Assuming the Central limit theorem, what is the correct representation of the sample means distributions of the random variables above?
Explanation:
Explanation
To determine the correct representation of the sample means distribution, we need to calculate the sample means and the covariance matrix.
Step 1: Calculate Sample Means
For X: Sum of X = 700 + 130 + 140 + 200 + 230 + 250 + 270 + 340 + 400 + 620 + 620 + 760 + 650 + 750 = 6,060 Sample size n = 14 Mean of X = 6,060 / 14 = 432.86 ✓
For Y: Sum of Y = 318 + 304 + 317 + 305 + 309 + 307 + 316 + 309 + 315 + 327 + 450 + 324 + 500 + 699 = 5,100 Mean of Y = 5,100 / 14 = 364.29 ✓
Step 2: Calculate Sample Variances and Covariance
Variance of X: Using the formula: Var(X) = Σ(x_i - μ_x)² / (n-1) After calculations: Var(X) ≈ 56,821.94 Sample variance for distribution of means = Var(X)/n = 56,821.94 / 14 ≈ 4,058.71 ✓
Variance of Y: Var(Y) = Σ(y_i - μ_y)² / (n-1) ≈ 12,807.34 Sample variance for distribution of means = Var(Y)/n = 12,807.34 / 14 ≈ 914.81 ✓
Covariance: Cov(X,Y) = Σ[(x_i - μ_x)(y_i - μ_y)] / (n-1) ≈ 15,489.18 Sample covariance for distribution of means = Cov(X,Y)/n = 15,489.18 / 14 ≈ 1,106.37 ✓
Step 3: Verify the Distribution
The correct representation is:
This matches Option C exactly. The covariance matrix is symmetric, so the off-diagonal elements should be the same (1,106.37), which is correctly shown in Option C.
Why other options are incorrect:
- Option A: Wrong Y mean (264.29) and wrong covariance matrix symmetry
- Option B: Wrong Y mean (464.29) and wrong covariance matrix symmetry
- Option D: Wrong Y mean (324.29) and wrong variance/covariance values