Question
Can someone please help me with this problem? Thank you in advance! 3. (10 points) Let...
Can someone please help me with this problem? Thank you in advance!
3. (10 points) Let X1, X2, ... be a sequence of random variables with common uniform distribution on (0,1). Also, let Zn = (11=1 X;)/n be the geometric mean of X1, X2, ..., Xn, n=1,2,.... Show that In , where c is some constant. Find c.
Answers
Answer! X ,X2) ~ Ulol) Zn = 1 {logy, logan - 1 Il-logxi) Zn = е "; = - logxin G (1) 2) I-logy, nacnol) syi an= YiNG(1) e let z = ΣΥ, SIN 2. G(0, 1) MGF og 20 2 és given by Mz(t) = (1-tin NOW E (Zn) Elet?} = (1-1-13h uning NIGA - n - (1+) (1+4) lim Elzn) = हा ol- e nyo liber (1+)" no lin E(Zn) = è lv (Zn) : E (Zo?) - (€ (2n))? +123): Ele?) (-2) (1+2) n 1 11 2n (1+2) (14+)? lim ulan) -13 (onus)") 2 17 lim 1 + +) lim hasol -2 O e Zn P, с cherot'c'will lim hyno E (2n) = el ēl C =
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