In each of the example below, assume you have a random sample {X1, X2, .... Xn) of size n. Answer the following questions:

Just need the solution steps to C1 and C2 the hints are just suggestions and don't necessarily need to be used.

Suppee if m Magi} with PDF fg[:; a} = $535. Tau are interested in eetimating the ppulatien variance 0'2. [Hate that the mean [if X is imam ten be equal tn zero, an there ie 1113 paint in estimating it}. :1. Show that the MLE of a? is given by sing = a; Is this the MOM eatinlaterl as well? Explain. e2. Shaw that ELLE veu feund in part :1 is unbiased1 eieient and eensis- tent. Hints: Thnd the MLE 01' Hg: The preblemie alittleless tedieusifyeu dene 95 Hi, I? :41. Then the PDF efa randem draw written in terms Di 3 is given by: frisgg} = me_;.hlasimize the leg-lilceliheed w.r.t. 3. Th shear unbiasedaess: Use the feet that 5121:?) = surng] fer each randem draw 'i [Why?}. 1Tb shear eieieney: War}: with fxlfmigd} and fellew the steps belew te nd the CHLB: 1] nd the leg-llkeliheed fer a random draw: :1 lizail = int frimii} = i1ca[2w} - thaw} - 515 [yen have this already} 2] difemntiate w.r.t. , i.e.nd the score: was = Easels a [you have this frem the F00 fer MLE} 3] calculate the varianee ef the snare: mania-5a} = wiaf} = = [shares = = at: *{EEIH - [Eimiilii' [using the denitien ei variance fer a variable 1'" = .5552}. Hew de we nd BEE}? The 4th mement ef a demeaned Nelmal randem variable is given by: Eili a)\" = He'd. In eur case p: = E]; hen-3e1 Eh = Bar" = 332. [Yen might remember that the hurteeis E{z,- pi)" ef the stande nermal withh=andei=lisj. Else we knew EEC?) = tr2 = 9. Th shear eensisteney: Use tha fact a is unbiased and shew that Ef[a} } E] as 11 :~ +ee. Saint-inn: IE Ii In erder te shew EFL-LE E dig-0M = L is efcient we need te shew it Tl: reaches CHLB fer unbiased estimaters ef :r