3
votes
0answers
52 views

Expected minimum of a finite random walk.

So I couldn't find any resource for how to calculate the expected minimum of a random walk. Since it is such the minimum of the random variables are actually not independent as they are cumulative ...
0
votes
1answer
77 views

Strong Law of Large numbers, prove expression is Standard Normal

Question: "Let $X_{1},X_{2},\cdots$ be a sequence of independent random variables such that $X_{n}$ is binomial with parameters $2n-1$ and $p=\frac{1}{2}$. If $$Y_{n} = \frac{2(X_{1}+X_{2}+\cdots ...
1
vote
2answers
57 views

Random walk confusion

If a ransom walk is binomial (1/2 probability of going forward, 1/2 backward) why isn;t the variance a) $\sigma=(\frac{n}{4})^.5$ b) instead of $\sigma=(n)^.5$ these sources seem to give ...
2
votes
0answers
58 views

Random walk type problem with time increments

Imagine you have $\$50$ and every $2$ minutes you either gain or lose $33$ cents. How would you model the evolution of the hypothetical bankroll for the next hour? My approach based on what i've read ...
1
vote
1answer
56 views

What's the name of this phemonenon in random walks?

Given a random walker on the number line that starts at 0 that has a 50% chance of going 1 unit in either direction every step, the walker will tend to stay on one side of the line for a while before ...
2
votes
1answer
65 views

Is there an unbiased random walk on a colored plane for any number of colors?

So I was trying to motivate the fundamental postulate of statistical mechanics (i.e. all microstates are assumed to be equally probable $-$ even if we can't practically measure them, but only their ...
3
votes
2answers
307 views

Expected value of function of random walk

I am trying to calculate $\lim_{n \to \infty} {E[e^{i \theta \frac{S_n}{n}}]}$. Where $\theta \in \mathbb{R}$, and $S_n$ is simple random walk. I could simplify it to $\lim_{n \to \infty}E[\cos(\theta ...
1
vote
2answers
618 views

Expectation of Random Walk

At each time step, I have 1/2 probability of walking one step to the right, and the same probability of walking one step to the left. Let X be the random variable corresponding to the final ...
2
votes
1answer
129 views

Expectation of $T^2$ where $T$ is the absorption time at ${a,−a}$ of a simple random walk $\{S_n\}$

I asked a very similar question before: Expectation of $TS_T$ where $T$ is the absorption time at $\{a,-a\}$ of a simple symmetric random walk $\{S_n\}$ But this time I have an ASYMMETRIC random ...
3
votes
1answer
309 views

Expectation of absorption time for a random walk which remains at n with probability 1/2

A random walk moves from k to k+1 with probability 1/2 and to k-1 with probability 1/2, except when k=n, in which case it remains at n with probability 1/2 and moves to n-1 with probability 1/2. ...