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3

The means add regardless of any other assumptions. Assuming (as usual in random walks) that the increments are independent, the variances also add. Hence the variance at time $t$ is $\sigma^2t$, and the mean is $\mu t$. Now $$\text{Var}(x_t)=E[x_t^2]-(E[x_t])^2=E[x_t^2]-\mu^2t^2=\sigma^2t$$ hence $$E[x_t^2]=\sigma^2t + \mu^2t^2.$$

1

Then after the mosquito goes to $0$, the mosquito must immediately go back to $1$ next move, i.e. the whole situation restarts. Consider the case that when the mosquito goes from $1$ to $4$ without passing $0$ as a success, and the case when the mosquito goes from $1$ to $0$ without passing $4$ as a failure. You already calculated the two probabilities as ...

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Let $P(i)$ denote the probability he'll get to $\$1,000,002$if he starts from$\$i$. We know that $$P(0)=0\;\;\;P(1,000,002)=1$$ You are asked to find $P(1,000,000)$. Imagine he has $\$i$and places a bet. He either gets to state$(i+1)$or to state$(i-1)$and we have: $$P(i) = \frac 13 P(i+1)+\frac 23 P(i-1)$$ It is easier to start near the "ruin ... 1 You haven't formalized what you mean by "the result is symmetric". The individual graphs are certainly not symmetric, you haven't specified a distribution whose symmetry we could inquire into, and if you had, it wouldn't be clear whether you could judge its symmetry by examining a couple of samples with the naked eye. So presumably what you mean is roughly ... 1 Well, I managed to hand-build a dirty distribution that works for me. The idea is to paste two geometric distributions together, and to truncate them not to get out of$S$: give yourself two parameters$k \in [0,\frac{2}{3}]$(overload for the probability of "staying here") and$\lambda \in \mathbb{R}^+$(parameter for the geometric distribution). compute ... 1 Your chain is recurrent (since it is a finite state makov chain). In this context there is a general formula that relates the invariant measure$\nu(\cdot)$and the expected time of return$\Bbb{E}[\tau_\cdot]$. Let$X_0 = i$, define$T_1 = \inf\{k>0, X_k = i\}, T_2 = \inf\{k> T_1, X_k = i\}, \ldots$Those$T_j\$ are the times of first return to the ...

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