# Question

If X and Y have a bivariate normal distribution, it can be shown that their joint moment– generating function (see Exercise 4.48 on page 139) is given by

Verify that

(a) The first partial derivative of this function with respect to t1 at t1 = 0 and t2 = 0 is µ1;

(b) The second partial derivative with respect to t1 at t1= 0 and t2 = 0 is σ21 + µ21;

(c) The second partial derivative with respect to t1 and t2 at t1 = 0 and t2 = 0 is ρσ1σ2 + µ1µ2.

Verify that

(a) The first partial derivative of this function with respect to t1 at t1 = 0 and t2 = 0 is µ1;

(b) The second partial derivative with respect to t1 at t1= 0 and t2 = 0 is σ21 + µ21;

(c) The second partial derivative with respect to t1 and t2 at t1 = 0 and t2 = 0 is ρσ1σ2 + µ1µ2.

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