Version:0.9 StartHTML:0000000105 EndHTML:0000008755 StartFragment:0000000141 EndFragment:0000008715 Suppose Y1 and Y2 are independent normal distribution with the same variance. Defifine U1 = Y1 + Y2,

U2 = Y1 Y2

a. Show U1 and U2 are joint normal

b. Show U1 and U2 are indenpendent

Hint:

If Y1 and Y2 are independent, they are joint normal such that (Y1, Y2) N2.

Use the results on page 6 and 8 from note 19: in order to show U1 and U2 are joint normal, you

need to show 1. their marginal distributions are both normal; 2. Any linear combination of U1 and

U2, cU1 + cU2 is still normal