Question

Let X₁, X₂ be two independent normal random variables with means μ₁, μ₂ and standard deviations σ1, σ₂, respectively. Consider Y=X₁−X₂, μ₁=μ₂=1, σ₁=1, σ₂=2. Then

A. Y is normally distributed with mean 0 and variance 1
B. Y is normally distributed with mean 0 and variance 5
C. Y has mean 0 and variance 5, but is NOT normally distributed
D. Y has mean 0 and variance 1, but is NOT normally distributed

Answer 1

The difference between two independent normal random variables is normally distributed with a mean equal to the difference of the means and a variance equal to the sum of the variances. So the correct answer is option B.

Step-by-step explanation:

The distribution of the difference between two independent normal random variables, Y = X₁ - X₂, is also a normal distribution with a mean equal to the difference of the means of X₁ and X₂, and a variance equal to the sum of the variances of X₁ and X₂. In this case, the mean of X₁ and X₂ is 1, and the standard deviations are 1 and 2 respectively. Therefore, the mean of Y is 1 - 1 = 0, and the variance of Y is 1 + 4 = 5. Therefore, the correct answer is B. Y is normally distributed with mean 0 and variance 5.