Question

Results for this suBHiSsIOH Entered Answer Preview Result (1/(1-b*t)*a correct Incorrect 26.25 26.25 Incorrect Incorrect 1.25 1.25 incorrect At least one of the answers above is NOT correct. 2 of the questions remain unanswered. (1 point) If Z is gamma(a, B), find the MGF of Z. (Enter a as a and B as b) M(t) = (1/(1-bt))*a Hint: If a = 20 and B = 4. differentiate the MGF you found above to find the first 3 moments of Z about 0 1st Moment: 2nd Moment 26.25 3rd Moment: Using the moments above, calculate the variance of Z. var(Z) = 1.25 Hint:

please provide detail.

Answer 1

solution - Given z n hamma (a, B) Plz) cere-48241 ; 270, 2/B 2/B 207 2-1 Bela ; zoo, a70, 820 MGF is Mz(t) = Eletz - Sage etz f (2) dz 1.5** *2 pata 248 24d pota Som E28 C1-8+) __ pola Estad ( since setzza'da Me(t) = (1 - BtJ"

Now, taking d = 20 and ß = 4 first moment is given Mi = E(2") as = { d at Mz(t)}t=0 = { d t ( 8 ) = { a (isto il restje *t p}} S t=0 3 = dB 20 x4 80 is given second moment by Me' = { 1 1 1 2 Meltsy - Para (T-Rea 1 Pc+T8 -4 XC ] - lat ape 0-pot yet I to =[AB Cati) (1-3 tja ( 11 t=a = a B (&+1)ß = a (x+1) 8² = 20821 x 4² = 6720 moment is given by Third My = { di ad M2 Lt)}

= { content (Pt" y tao * C4+1) Bt Citadele jt- - paca+l) $${-(6+2)} (-8t) EBY tro Variance = a (x+1) (2+2) B3 20 X 21 X 22 X 43 = 591360 of z is given as M₂ = l,' -", = 6720 - 80² =6720 - 6400 320