Please help and please show your work 650 Let Z1 and Z2 be i

Please help and please show your work

6.50. Let Z1 and Z2 be independent standard normal random variables. Show that X, Y has a bivariate normal distribution when X = Z1, Y = Z1 + Z2.

Solution

z1 is normal with N(mu1, sigma1)

Z2 is normal with N(mu2, sigma2)

Z1+Z2 has mean mu1+mu2 and variance sigma1^2+sigma2^2

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As both x and y are normal X,Y has a bivariate normal distribution.