Question

Let \( (X_1, Y_1), (X_2, Y_2), \ldots, (X_{20}, Y_{20}) \) be a random sample from the \( N_2(0, 0, 1, 1, \frac{3}{4}) \) distribution. Define
\[\bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad \bar{Y} = \frac{1}{20} \sum_{i=1}^{20} Y_i.\]
Then \( \text{Var}(\bar{X} - \bar{Y}) \) is equal to:

• A. \( \frac{1}{16} \)
• B. \( \frac{1}{40} \)
• C. \( \frac{1}{10} \)
• D. \( \frac{3}{40} \)

Answer 1

The Correct Option is B

1. Variance of \( \bar{X} \): - Since \( X_i \sim N(0, 1) \), the variance of the sample mean \( \bar{X} \) is: \[ \mathrm{Var}(\bar{X}) = \frac{\mathrm{Var}(X_i)}{n} = \frac{1}{20}. \] 2. Variance of \( \bar{Y} \): - Similarly, since \( Y_i \sim N(0, 1) \), the variance of the sample mean \( \bar{Y} \) is: \[ \mathrm{Var}(\bar{Y}) = \frac{\mathrm{Var}(Y_i)}{n} = \frac{1}{20}. \] 3. Covariance of \( \bar{X} \) and \( \bar{Y} \): - The covariance between \( X \) and \( Y \) is given as \( \frac{3}{4} \), so: \[ \mathrm{Cov}(\bar{X}, \bar{Y}) = \frac{\mathrm{Cov}(X, Y)}{n} = \frac{\frac{3}{4}}{20} = \frac{3}{80}. \] 4. Variance of \( \bar{X} - \bar{Y} \): - Using the formula for the variance of a difference: \[ \mathrm{Var}(\bar{X} - \bar{Y}) = \mathrm{Var}(\bar{X}) + \mathrm{Var}(\bar{Y}) - 2\mathrm{Cov}(\bar{X}, \bar{Y}). \] - Substituting values: \[ \mathrm{Var}(\bar{X} - \bar{Y}) = \frac{1}{20} + \frac{1}{20} - 2\cdot\frac{3}{80} = \frac{2}{20} - \frac{6}{80} = \frac{8}{80} - \frac{6}{80} = \frac{2}{80} = \frac{1}{40}. \]