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I have the following problem based on the random walker. I have no idea what steps I should take to solve this exercise. If you could solve it and explain each step to me it would be very helpful. Thank you!!!

A drunken person walk along a line, making steps sized 1. Each step is either to the right or the left, and the probability to walk to the right is fixed: $P(right) = p$. Let $X_{n}$ be the position of the point after $n$ steps. Find $E(X_{n})$ and $V(X_{n})$

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We will assume each step is of length one. Denote $W_1, W_2, \cdots, W_n$ such that each $W_i$ states the direction of the person taken at time $i$, specifically $$W_i=\begin{cases}1&\text{ probability }p,\\ -1&\text{ probability }1-p.\end{cases}$$ Therefore, $$E[W_i]=p(1)+(1-p)(-1)=2p-1.$$ From here we know if $p>1/2$ then the person is expected to walk to the right, and vice versa. Moreover, $$Var(W_i)=E[W_i^2]-E[W_i]^2=1-(2p-1)^2=4p(1-p).$$ Note that $W_i^2$ is always $1$.

Using the relation $X_n=\sum_{i=1}^n W_i$, we have $$E[X_n]=E[W_1]+\cdots+E[W_n]=n(2p-1),$$ and since it is obvious each step is independent from other steps, we have $Cov(W_i,W_j)=0$ if $i\neq j$, hence $$Var(X_n)=Var(W_1+\cdots+W_n)=Var(W_1)+\cdots+Var(W_n)=4np(1-p).$$

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    $\begingroup$ I have a question, why can't I do this $E[W_{i}]=p(1)+(1−p)(−1)=2p−1$ with $E[X_{n}]$? $\endgroup$
    – Alex100
    Commented Jan 16, 2021 at 10:26
  • $\begingroup$ It will be harder and harder to keep track of $X_n$ with $n$ large. Imagine $X_2$, then there are three cases for us: $X_2$ can equal to $-2,0,2$. Where $P\{X_2=2\}=p^2, P\{X_2=0\}=2p(1-p)$ and $P\{X_2=-2\}=(1-p)^2$. If we directly calculate the expected value then it will be a massive mess when $n$ is large. We use $W_i$ because it will be easier to work with. $\endgroup$
    – kelvin hong 方
    Commented Jan 16, 2021 at 10:31
  • $\begingroup$ Thank you very much @kelvin hong 方, you were very helpful!!! $\endgroup$
    – Alex100
    Commented Jan 17, 2021 at 8:07
  • $\begingroup$ You're welcome @Alex100 $\endgroup$
    – kelvin hong 方
    Commented Jan 17, 2021 at 10:17

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