Exercise 3:Let (Yn)nEN be a sequence of random variables with the property -a < Yı < 0 for a constanta, 0 < a < ∞, and Ym < Yn for Vm < n.i) Why does the Monotone Convergence Theorem for (Y,)nEN not hold?ii) Show thatlim E[Yn] = E[ lim Y,].

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Answered: Exercise 3: Let (Yn)neN be a sequence…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 34E

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Theorem: Suppose (Xn)n≥1 is a sequence of random variables with corresponding momentgenerating functions M_Xn , and X is a random variable with moment generating functionM_X such that for some δ > 0 we have M_X (t) < ∞ for all t ∈ (−δ, δ). If lim n→∞ MXn (t) = MX (t) for all t, then lim n→∞ F_Xn (x) = F_X (x) for all x where F_X is continuous. That is, if the moment generating functions of X_n converge to the moment generating function of X, then the distribution of X_n converges to the distribution of X. Use this to show that if Sn ∼ Binomial(n, λ/n ), then the distribution of Sn converges to Poisson(λ) as n → ∞.
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Exercise 3: Let (Yn)neN be a sequence of random variables with the property -a < Y, < 0 for a constant a, 0 < a < 0, and Ym < Yn for Vm < n. i) Why does the Monotone Convergence Theorem for (Y)nEN not hold? ii) Show that lim E[Y,] = E[ lim Y,]. n-00