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1.1 [Probability and Statistics] Let X and Y be jointly distributed normal random variables, where cov[X, Y]-2 In other words, the joint distribution of the pair (X, Y) ~N(,),where 1 |.and Σ := |.-2 9 What is the distribution of the random variable Z:-X -2Y?

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E(X -2Y) E(X)-2E(Y)12 3, Var(X-2Y) = Var(X) + 4Var(y)-4Cor(x, y) = 1 + 4 * 9 + 8 = 45 We know, if (X, Y) follows Bivariate normal distribution then the linear form of X andY i.e. aX +bY follows univariate normal distribution with mean-E(aX+ by) and Variance = Var(aX + bY) Hence. Z = X-2Y ~ Ņormal(3.45)

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