Answer to Question #320991 in Statistics and Probability for Amos

Question

The moment generating function of two jointly distributed random variables X1 and X2 is M(t1, t2) = e ^− 0.5 G where G = (7.51t 2 1 + 7.9t 2 2 + 3.8574t1t2 + 135.4t1 + 137.2t2) Using this function, find the correlation coefficient of of X1 and X2

Answer 1

$M\left( t_1,t_2 \right) =\exp \left( -\frac{7.51{t_1}^2+7.9{t_2}^2+3.8574t_1t_2+135.4t_1+137.2t_2}{2} \right) \\EX_1=\frac{\partial M\left( t_1,t_2 \right)}{\partial t_1}=-\frac{135.4}{2}=-67.7\\{EX_1}^2=\frac{\partial ^2M\left( t_1,t_2 \right)}{\partial t_1^2}|_{t_1=t_2=0}=4575.78\\Var\left( X_1 \right) =4575.78-\left( -67.7 \right) ^2=-7.51<0\\Since\,\,we\,\,obtained\,\,a\,\,negative\,\,variance, the\,\,problem\,\,is\,\,incorrect$