4. Let 6>0. Let X1, X2, ..., Xn be a random sample from the distribution with probability density function f ( x; 8) = 63 240 x2 e - Vox x>0, zero otherwise. a) (i) Obtain the maximum likelihood estimator for 6, 8. That is, find 8 = arg max L(6) = arg max In L(8), where L(8) = IIf(x; ;6). i=1 (ii) Suppose n=3, and X1 =2.25, X2=5.76, x3 = 10.89. Obtain the maximum likelihood estimate for 8, 8. b) Show that W = VX has a Gamma distribution. What are its parameters of and 0? c) n=3 Suppose n=3, and 6=6.76. Find the probability P Z X; 25. 1= 1 d) Find a closed-form expression for E(X*), k>-3. Hint: X = W- and If T has a Gamma(a, 0= /, ) distribution, then E(TM ) = 0" r(atm) _ r(a+m) r(a) am r ( a) m > - a. OR u = vox and r(a)= [ud-le-" du, a>0, e) (i) Obtain a method of moments estimator for 6, 8. That is, if E(X) = h(8), solve X =h( 6 ) for 6 . (ii) Suppose n =3, and x =2.25, x2 =5.76, X3 = 10.89. Obtain a method of moments estimate for 8, 8