Question

3. Let Yİ ~Gamma ( -3,ß-3), Y ~Gamma( -5, ß-1), and W-2% + 6K. a) (9 pts) Find the moment generating function of W. Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and parameters)?

Answer 1

as mgf of gamma distribution M_x(t) =1/(1-$\beta$t)^$\alpha$

hence mgf of Y1 =M_y1(t)=1/(1-3t)³

therefore mgf of 2Y1 =M_2y1(t) =M_y1(2t)=1/(1-3*2t)³ =1/(1-6t)³

similarly mgf of 6Y2=M_6y2(t) =1/(1-1*6t)⁵ =1/(1-6t)⁵

hence mgf of W =M_w(t)=M_2y1(t)*M_6y2(t) =(1/(1-6t)³)*(1/(1-6t)⁵) =1/(1-6t)⁸    t<1/6

b)

comparing mgf of W with that of gamma distribution of M_x(t) =1/(1-$\beta$t)^$\alpha$

W follows gamma distribution with parmeter $\alpha$=8 and $\beta$=6