Question

Suppose x1∼ N(1, 5) and x2 ∼ N(−2, 2). The covariance between x1 and x2 is 1.3.

What is the distribution of x1+ x2? What is the distribution of x1− x2?

Answer 1

The student's question concerns the distribution of the sum and difference of two normally distributed variables, x1 and x2. The sum x1 + x2 is N(-1, 9.6) and the difference x1 - x2 is N(3, 5.4).

Step-by-step explanation:

The student is asking about the distribution of the sum (x1 + x2) and the difference (x1 - x2) of two normally distributed random variables x1 and x2. When variables are normally distributed and we know their means, variances, and covariance, we can find the distribution of their sum and difference. The sum of two normally distributed random variables is also normally distributed with a mean equal to the sum of their means and a variance equal to the sum of their variances plus twice the covariance. Therefore, x1 + x2 has a normal distribution with mean 1 + (-2) = -1 and variance 5 + 2 + 2(1.3) = 9.6. The difference, x1 - x2, has a normal distribution with mean 1 - (-2) = 3 and variance 5 + 2 - 2(1.3) = 5.4.

Accordingly, x1 + x2 is normally distributed as N(-1, 9.6), and x1 - x2 is normally distributed as N(3, 5.4).