48 Example of a nonnormal bivariate distribution with normal

4.8. (Example of a nonnormal bivariate distribution with normal marginals) Let X1 be N(0, 1), and let X2 = { Show each of the following. (a) X2 also has an N(0, 1) distribution. (b) X1 and X2 do not have a bivariate normal distribution Hint: (a) Since which is a standard normal probability. (b) Consider the linear combination X1 - X2, which equals zero with probability P [ |X1| > 1] = .3174.

Solution

X1 : N(0,1)

X2: N(0,1)

But we must prove that x1, x2 do not have a bivariate normal distribution though they are normal separately.

As both X1 and X2 are std normal variates X2 has a N(0,1) distribution.

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b) Consider x1-x2 =y

y has value as 0

Mean =0

P(|X1|>1 ) =0.3174 which contradicts the fact x1-x2 is normal

Hence x1-x2 is not normal though x1 and x2 are normal.