Question

4. Let UX and V~xbe independent, and s > 2. i) Compute E (H] ii) Suppose T has a t-distribution with degree of freedom s > 2. Compute E[T) and Var(T). (Hint: use i) to find E [T?]).

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Answer 1

Un ras, Vaux} be independent mv. ~ Flr,s) - distribution with Paf is given by and se, by definition, ull know that 22-y!!!!!!! 2012: Criven that: Ulr vis 712 쓸 x ) 912 ( x fins = P ( 2 ) (i+ 5x) " Then, E(X) = ? Elx) = (8) ida h ( 7 ) 2 712 xflag de a 2 !! set 3 Then. 3 en to y. dx = d.dy. (5) "92.5" (5.4); (f.y jf P ( 2 ) E14) = (1+y)(4) P 12;ß), fly) 2 y lt 2 - d1 dy RCC) I+AS-1 y ő (1+y) 1 lit yg aku olylo () ß ( 2 ) B (nem) Here & = /211, 8 = 8/21 = (story (sim). . M2 (B/ litriz (S/21 hinta C-CENT siz olxista Šaz 11 52 z Sleut CS Scanned with CamScanner

Lootcoo ） I t dt Tht-dut with S dof. $72. that t-dist 18 symmetric about line t=0, then. Then pdf of a is f1t) gle Then suppose JLH) - t. Elt) =1 Bl*, ) (1***J** ELT) = 5 +.f14). dt = steplaneje (154 is add function, and we know $+1 2 glth.dt glt)adt a (: 0 tro tco 80 1 E(T)-0 By definition, we know that z Z NM10,12 T= Ats geen mens independent. 2 z 2²1; Nflis) - By definitions OR T2 menys 12 ass Then use result &, we get. E(T) = E(X) = S S-2 Then. var(t) = € (72) - (E (7) J? S Sa -OR war (9) S S-2 Any CS Scanned with CamScanner