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.)


Probability
Ian Dumais Ian Dumais
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Nirenberg Nirenberg
  • Ian Dumais Ian Dumais

    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?

  • Ian Dumais Ian Dumais

    Of course. I had an error in my notes. Thanks again!

  • Nirenberg Nirenberg

    F_W_{n+1} is the cdf of W_{n+1}. It's derivative will give you the pdf of W_{n+1}.

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