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10. Suppose (Zk) are iid mean 0 random variables with EZ2 = 1. Sup- pose that for each m € N, Nm is a Poisson (m) random variable inde- pendent of all (Zk). Prove that Nm Σ Zk k=1 (d) m-x → N(0, 1). Hint: one method to do this goes by computing the characteristic func- tion using the law of total expectation the moment generating func- tion of a Poisson random variable which you can find on Wikipedia might be helpful - and calculus, like we did for the proof in class of the central limit theorem. You may assume for simplicity that the characteristic function of Z₁ is real-valued.

10. Suppose (Zk) are iid mean 0 random variables with EZ2 = 1. Sup- pose that for each m € N, Nm is a Poisson (m) random variable inde- pendent of all (Zk). Prove that Nm Σ Zk k=1 (d) m-x → N(0, 1). Hint: one method to do this goes by computing the characteristic func- tion using the law of total expectation the moment generating func- tion of a Poisson random variable which you can find on Wikipedia might be helpful - and calculus, like we did for the proof in class of the central limit theorem. You may assume for simplicity that the characteristic function of Z₁ is real-valued.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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10. Suppose (Zk) are iid mean 0 random variables with EZ2 = 1. Sup- pose that for each m € N, Nm is a Poisson (m) random variable inde- pendent of all (Zk). Prove that 1 'm Nm k=1 Zk (d) → N(0, 1). 8个W Hint: one method to do this goes by computing the characteristic func- tion using the law of total expectation - the moment generating func- tion of a Poisson random variable which you can find on Wikipedia might be helpful - and calculus, like we did for the proof in class of the central limit theorem. You may assume for simplicity that the characteristic function of Z₁ is real-valued.
Transcribed Image Text:10. Suppose (Zk) are iid mean 0 random variables with EZ2 = 1. Sup- pose that for each m € N, Nm is a Poisson (m) random variable inde- pendent of all (Zk). Prove that 1 'm Nm k=1 Zk (d) → N(0, 1). 8个W Hint: one method to do this goes by computing the characteristic func- tion using the law of total expectation - the moment generating func- tion of a Poisson random variable which you can find on Wikipedia might be helpful - and calculus, like we did for the proof in class of the central limit theorem. You may assume for simplicity that the characteristic function of Z₁ is real-valued.
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