Question

(a) Suppose X1 ∼ N(0, 2) and X2 ∼ N(10, 8) are normal variables with correlation −0.5. What is the mean and variance of Y = 2X1 − X2? Hence or otherwise, state the distribution of Y .

Answer 1

The mean of Y, found using the linear combination of the means of X1 and X2, is -10. The variance of Y, found using the linear combination of variances and considering the covariance, is 32. Hence, Y follows a normal distribution, specifically N(-10, 32).

Step-by-step explanation:

To find the mean and variance of Y = 2X1 - X2 where X1 ∼ N(0, 2) and X2 ∼ N(10, 8), we use the properties of linear combinations of normal variables. For the mean of Y, we simply take the linear combination of the means:

μ_Y = 2*μ_X1 - μ_X2 = 2*0 - 10 = -10.

For the variance of Y, since X1 and X2 are correlated, we also need to take into account the covariance. The variance given by:

σ_Y^2 = 2^2*σ_X1^2 - 2*2*-0.5*σ_X1*σ_X2 + σ_X2^2 = 4*2 - 2*2*(-0.5)*√2*√8 + 8 = 16 + 8 + 8 = 32.

Therefore, the distribution of Y is also normal since it is a linear combination of two normal variables, so Y follows a normal distribution with mean -10 and variance 32, or Y ∼ N(-10, 32).