Find Mean, Variance, and Distribution of Recursively Defined Sequence of Random Variables
The sequence of random variables $\{W_n\} (n=1,2,...)$ is defined recursively by the formula
$$W_{n+1}=B_{n+1}W_n+Z_{n+1}, n\ge1, W_1=Z_1$$ where $\{Z_n\}$ are IID standard normal random variables, and $\{B_n\}$ are IID random variables with distribution
$$P(B=1)=p, P(B=-1)=1-p.$$ Assuming that $\{Z_n\}$ and $\{B_n\}$ are independent, find the mean and variance of $W_n$ for all $n\ge1$, and find the distribution of $W_n$ (either pdf or cdf and parameters of standeard distribution.)

58
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment

4.8K
-
Thanks! I don't think it affects the overall logic of the proof, but wouldn't F_W_{n+1}, if that is the cdf, be equal to the derivative of the stated probability?
-
Of course. I had an error in my notes. Thanks again!
-
F_W_{n+1} is the cdf of W_{n+1}. It's derivative will give you the pdf of W_{n+1}.
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 883 views
- $15.00
Related Questions
- Three red balls and three green balls on a circle
- Exponential Probability
- Coincidence or pattern?
- Questions for Statistics Project
- Convergence in Lp
- Five married couples are seated around a table at random. Let $X$ be the number of wives who sit next to their husbands. What is the expectation of $X$ $(E[X])$?
- Three unbiased coins are tossed. What is the probability of getting at most two heads?
- Exponential random variables