Question

9. In many cases where a sequence of random variables converges in probability to some b, this b will be either the expected value or the limit of the expected values of the variables. However, this is not generally true. (a) Consider a sequence of random variables where for each n, xn comes from this distribution with P(Xn = n) = 1/n and P(Xn = 0) = 1 - 1/n. Find limn+ E(Xn). (b) Find the value b such that Xn ? b. (C) Now, consider a sequence of random variables where for each n, Xn comes from this distribution with P(Xn = n2) = 1/n and P(Xn = 0) = 1-1/n. Find limnto. E(Xn) and the value b such that Xn ? b. n TU n

Answer 1

Ca PC xn an)a and p C Xn=0)=1 E (xn )= nx tox (-) ノナ0 = lim ECXm)=) 1. %3D distribution fumiton Now, the F. Cxa)= ーn, oく2入れ lim Fxn Cn) = is the distribubion Lunchin ofa deg enerade random variable at the point Xn =7 Ex only if a degenerale rand em variable if -) Xn Xn =-> where

-> Here -> C) E Cx )2 n°xt + ox( →) n to dim E(x) = Now (x)= Xn 1-, os%<n? Now lim (x)S0, x<O -> > Xn 20 => Xn Xn b -7 b= 0 9. which is dlegonenak at o: