Let (Xn} be a sequence of rv's such that X, is distributed as Z with probability pn and as onZ with probability 1- Pn where Z~ N(0, 1), pn →pe [0,1] and on→ o. |(a) Show that X, X if and only if p =1. What is the distribution of X? |(b) lim, Var(X„) is not necessarily the same as Var[X]. Find the possible values of lim, Var[X,).

MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Let (X,} be a sequence of rv's such that X, is distributed as Z with probability pPn
and as onZ with probability 1 – Pn where Z- N(0, 1), pn →pe [0, 1] and on → o.
|(a) Show that X, x if and only if p = 1. What is the distribution of X?
(b) lim,- Var[X„] is not necessarily the same as Var[X]. Find the possible values
of lim,- Var[X,).
Transcribed Image Text:Let (X,} be a sequence of rv's such that X, is distributed as Z with probability pPn and as onZ with probability 1 – Pn where Z- N(0, 1), pn →pe [0, 1] and on → o. |(a) Show that X, x if and only if p = 1. What is the distribution of X? (b) lim,- Var[X„] is not necessarily the same as Var[X]. Find the possible values of lim,- Var[X,).
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