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1. Let X and Y be jointly normal with means 0, variances 1, and correlation coefficient ρ. Compute the moment generating function of X · Y for (a) ρ = 0, and (b) general ρ.

2. Suppose X1, X2, and X3 are independent and N (0, 1)-distributed. Compute the moment generating function of Y = X1X2 + X1X3 + X2X3.

3. If X and Y are independent, N (0, 1)-distributed random variables, then X2 + Y 2 ∈ χ2(2) (recall Exercise 3.3.6). Now, let X and Y be jointly normal with means 0, variances 1, and correlation coefficient ρ. In this case X2 + Y 2 has a noncentral χ2(2)-distribution. Determine the moment generating function of that distribution.

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