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Results tagged with stochastic-processes
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user 48831
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
1
vote
Dice waiting time for $5,6$ is smaller than for $6,6$
Here is a completely elementary argument.
Let $T_{56}:=\tau_{56}+1, T_{66}:=\tau_{66}+1$ be the first times
the patterns have appeared, i.e.
$$T_{56}:=\inf\{ n \geq 2\;:\;X_{n-1}=5,\,X_n=6 \}$$ etc. …
- 3,255
1
vote
Minimizer of two random walks
The topic belongs to the fluctuation theory
of random walks. See e.g. Feller II, 2nd ed. (1970), chapter XII.
Here is some useful (condensed) information:
(1) (for each $n$) the position of the (fi …
- 3,255
0
votes
The min of the mean of iid exponential variables
(Since you are looking for a reference, I turn my comment above into an answer:)
A proof using classical fluctuation theory is given my answer to
Expected supremum of average?
(I'm not aware that this …
- 3,255
3
votes
Accepted
How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\si...
Call the left resp. right hand sum $R_m$ resp. $R_{m+1}$. As $L$ is $\exp(\lambda)$ and independent of $(R_m,R_{m+1})$ , taking expection with resp. to $L$ first gives
$$\mathbb{P}(R_m\leq L < R_{m+1 …
- 3,255