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Suppose X is a N(0, 1) and Y is a N(1, 2). They have a cov[x, y] = 2. What is the distribution of (2*X - 3*Y).

I was thinking since they are not independent, any linear combination of them would not result in a normal distribution. But, would any combination of them produce Z a bivariate Normal?

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You have missed an essential information: are X and Y jointly normal? –  leonbloy Oct 15 '12 at 23:57
    
We don't know if they are jointly distributed, the question asks if 2X - 3Y has a normal distribution. –  Josh Oct 16 '12 at 0:01
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Your second paragraph is wrong (or at least bad expressed) on two accounts: 1) the linear combination of two jointly normal variables (independent or not) is normal (otherwise you cant' say anything) 2) Z=2X - 3Y cannot be a "bivariate" normal, if X and Y are scalars Z is also a scalar –  leonbloy Oct 16 '12 at 0:10
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