3 votes
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Simple solution to random walk

One way to solve this problem is to represent the possible courses of the match by paths through a directed graph. Each vertex of the graph represents a possible score $\ (h,a)\ $ that might have ...
lonza leggiera's user avatar
3 votes

Simple solution to random walk

With the $X$-axis representing score of the home team and the $Y$-axis that of the away team. write down scores as $0-3\mid 1-3 \mid 2-3 \mid 3-3 \mid 4-3$ $0-2\mid 1-2 \mid 2-2 \mid 3-2 \mid 4-2$ $0-...
true blue anil's user avatar
3 votes
Accepted

Why does the Wiener process use $\sqrt{dt}$ instead of $dt$? Why does simulation of random walk in continuous-time not occur as expected?

Essentially because the variance scales quadratically, e.g. $\operatorname{Var}(tX) = t^2\operatorname{Var}(X)$. I'm only going to talk about discrete time processes, rather than something like ...
daisies's user avatar
  • 1,548
2 votes
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The stationary distribution of a rook's random walk

There are a few ways to do this, the most straightforward probably being by a spectral argument. For that, you may be interested in a theorem from (1): Theorem. For an irreducible and aperiodic ...
Matthew Sutter's user avatar
2 votes
Accepted

Summation notation in an expectation formula

In this context, I interpret $\sum_{1\le m\le n}$ as being a double sum over two indices $m$ and $n$. I don't love that notation though. I would write this calculation as $$ \mathbb{E}(N) = \sum_{n=1}^...
Greg Martin's user avatar
  • 78.9k
2 votes
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Asymptotic form of solution to biased random walk

I think that you skipped some steps when doing you did the approximations in the unbiased case. If you do it rigorously, the same method applies. More formally, you want to estimate large deviations ...
LPZ's user avatar
  • 2,948
1 vote
Accepted

Difference equations applied to Random Walk

Let $p^*=p/(p+q)$ and $q^*=q/(p+q)=1-p^*$. Then if you consider your walk only at times when it does not stay put, it's a classical [Gambler ruin problem][1]. The $p_k$s are, of course, the same for ...
van der Wolf's user avatar
  • 2,327
1 vote
Accepted

Why is the Inner Product Induced by a Gaussian Matrix a Gaussian Process?

With your notation $X_{uv}$ is a linear combination of $A_{ij}$, namely $X_{uv}=\sum_{1\le i\le m,1\le j\le n}v_iu_jA_{ij}$. So $$ \sum_{(u,v)\in T_0}a_{uv}X_{uv}=\sum_{(u,v)\in T_0}\sum_{1\le i\le m,...
Will's user avatar
  • 6,927
1 vote

Random walk where Increments have exponential distribution. Probability of never reaching a negative value after $n$ steps.

This is not a full answer. I found a different perspective to view the problem in, which turns the question from a complicated integral to a complicated discrete summation. Consider a Poisson process,...
Mike Earnest's user avatar

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