Question

3. Let N = R and P be the probability distribution on (R, B(R)) with density 1 XER. Put X(w):=w2, WEN. (a) Describe o(X) (the o-algebra on 2 generated by X). Justify your answer. (b) Derive the distribution function Fx of the random variable X. (c) Compute the mean EX and moment generating function y(t) := t> 0, for the random variable (RV) X. [3+3+3=9] EetX 300,

Answer 1

(a) The 6-algebala ((.)) is Set Subsety-R- 3) Solution Let us Consider 2 = R and pbe distrübution eoitti density -P) - lxl probability On (R, BLR)) XER Such that for w. eo(x) => woon(x) (6) het y follows double Exponential distribution with parameter Here x=y J = Jacobran ! - pdf of x fx (0) x20 &0, otherwise cdg of Fx (2)=0 ,u<0 and S. fort)dt, =0,450 $ get * e you Stdt, azo e ЧЈЕ 50,250.& é Zdt, x20, 0 Now Substitute z=ſic Then,

= 0,850, & l-ev), se 2 U20 (C) E(X) = $ te vedt 407 s yee redt Le izrezat Substitute z=si з, Elett). So you eta beste dz, retana Again substitute 2 -(FZ- are laz

Cm Let (2) S Set(2-aayat 놀한 ㅗ Vât (2- d2 ENTRE 12t (iso() elstelsint at 글e 았 아 ent(a . GE 이 -) e 화 는 ((0) - (-)) (1 - ( )) Hence Proved Nat