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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 11 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON