Problem 1 Let X ~ [(a, B) for a > 0, 8 > 0. Recall X has support on [0, co) and pdf given by 70-1 -x/B f(I) = Ber(a) where T(a) = (a) Show [(a + 1) = of (a). (Hint: integration by parts) (b) Compute D(1), I() and T(). NB: Fully justify your steps! For example, if you want to use Euler's reflection formula, then you need to prove Euler's reflection formula. Also, you don't need to use Euler's reflection formula at all here. For instance, you can establish a relationship between the Gamma function when a = = and a Gaussian integral through a u-substitution, and you can use properties of Gaussian integrals. (c) Compute EX and Var X. (d) Compute the mgf Mx (t) = Eetx.W yum ( d B ) for do /P /Q 10 on Lo, do ) pdf : f (x ) = x2-1p-x/P Bar ( d ) To where m( d ) - sxe edx ( o a show n ( d + 1 ) = of (d ) Hint : integration by parts 9 ( d + 1 ) = s x ( * +1) - e - " dx = 5 y " e = * d x 7 00 9 a ( d ) = 2 5 x a - 919 u = ya+1 ->du /x = ( + 1 )x 15e- 27 , du = - ( d+1 ) [ ( a tl , u atm ? -FI Seat du = - 1 ( d+ 1, 4at1)= -1 ( d+ 1 / x ) So, Syde " d x = - [ ( d + 1 x) th sayd e - x d y = [ ( d + 1 ) , a + 1 >0 xe dx an ( d = d s x de d x I'm having 272 ) = - [( a , x ) fromble s x e d x = - d [ a X ( ) ? 5 x d - edx = al (x ) , a zo following 90 as ( " x "' e x d x = d ) x e d x Rewrite int . by parts using u = x dve d x for [karl) - ). xe dxr( ) 25 x 2edx= 5x = ( ( v ? ) - 2ezudy = 25e du For Gaussian density if M= 0 62 : 72 - 00 du - 00 2 5 e du =Vit CH ( X = xf ( x dy= (x x ( d x -00 E( x ? ) = / 2 4 5X 2 X - F 7 2 B 2 ( 2 + 1) / 50 xxe-XP D var (x- BUG -1( X ) 7 d ) M x ( 4 ) - F ( p t x\\= 5 0 + - * / d X You can use a a/10 u-sub to get the intermal in nice form you a can compute