Added some solutions

[Edlingeer]

[DanielSank]

Thank you! I might not get around to this until the weekend, but in any case the contribution is very much appreciated.

[DanielSank]

I'm finally returning to this project. Yay. Should have this merged soon.

[Edlingeer]

Hi,

No problem, I'll fix this after work today. And thanks for the feedback.

-- Thomas

-- Thomas

2017-03-06 9:39 GMT+01:00 Daniel Sank [email protected]:

@DanielSank requested changes on this pull request.

Thanks, this is great. There are a couple of little punctuation errors. If you can fix them up I'll merge this.

In chapter2/Description_sequence_random_events.tex https://github.com/DanielSank/vankampen-stochastic/pull/10#discussion_r104363415 :

@@ -0,0 +1,24 @@ +\leveldown{Description of a sequence of random events - pg. 32}

+\leveldown{Problem} + +Calculate $Q_s$ for the given recipe of construction of a random set of dots +\levelstay{Solution} +We need to find a description of a sequence of random events at $\tau_1$, $\tau_2$, ... in terms of the probability distribution $Q_s$ using the probability $w$. The $w$ probability is the simplest description of correlated random events : the new arrival is dependent on the timing from the previous arrival and both arrivals are governed by the same probability function $w$. We may as well represent this situation by having $s$ replicas of the system and reporting the first random events in each of these systems.Such first event would be governed by a probability $w$. We can write $Q_s$ as a probability of events that are independent of one another in each of these systems.

There are a few punctuation errors here, i.e. a space missing at the end of a sentence, etc.

In chapter2/Mean_square_N_interval.tex https://github.com/DanielSank/vankampen-stochastic/pull/10#discussion_r104363609 :

+\begin{equation} +\begin{split} +& \int_{-\infty}^{\infty}\chi^2(\tau_i)Q_s(\tau_1,\tau_2,...,\tau_i,...,\tau_s)d\tau_1d\tau_2...d\tau_s\ +& \Leftrightarrow \int_{-\infty}^{\infty}\chi^2(\tau_i)Q_s(\tau_i,\tau_2,...,\tau_1,...,\tau_s)d\tau_1d\tau_2...d\tau_s \ +& \Leftrightarrow \int_{-\infty}^{\infty}\chi^2(\tau_1)Q_s(\tau_1,\tau_2,...,\tau_i,...,\tau_s)d\tau_id\tau_2...d\tau_1...d\tau_s \ +& \Leftrightarrow\int_{t_a}^{t_b}\left[d\tau_1 \int_{-\infty}^{\infty}Q_s(\tau_1,...,\tau_s)d\tau_2...d\tau_s \right] +\end{split} +\end{equation} +in the same fashion we can transform the expression $(b)$ getting the following result +\begin{equation} +\begin{split} +& \int_{-\infty}^{\infty}\chi(\tau_i)\chi(\tau_j)Q_s(\tau_1,\tau_2,...,\tau_s)d\tau_1d\tau_2...d\tau_s \ +& \Leftrightarrow \int_{t_a}^{t_b}\left[d\tau_1\int_{t_a}^{t_b}d\tau_2\int_{-\infty}^{\infty}Q_s(\tau_1,\tau_2,...,\tau_s)d\tau_3...d\tau_s\right] \ +\end{split} +\end{equation} +in the end we can rewrite $\langle N^2 \rangle$

Capitalize beginning of new sentence?

In chapter2/Description_sequence_random_events.tex https://github.com/DanielSank/vankampen-stochastic/pull/10#discussion_r104363908 :

+\leveldown{Problem}

+Calculate $Q_s$ for the given recipe of construction of a random set of dots +\levelstay{Solution} +We need to find a description of a sequence of random events at $\tau_1$, $\tau_2$, ... in terms of the probability distribution $Q_s$ using the probability $w$. The $w$ probability is the simplest description of correlated random events : the new arrival is dependent on the timing from the previous arrival and both arrivals are governed by the same probability function $w$. We may as well represent this situation by having $s$ replicas of the system and reporting the first random events in each of these systems.Such first event would be governed by a probability $w$. We can write $Q_s$ as a probability of events that are independent of one another in each of these systems. +\begin{equation} +Q_s(\tau_1, \tau_2,...,\tau_s)=P(\tau_1)P(\tau_2)...P(\tau_s) +\end{equation} +Thus we have +\begin{equation} +\begin{split} +& P(\tau_1)= w(\tau_1) \ +& P(\tau_2)= \int_{0}^{\tau_2}w(\tau_2-\tau_1)d\tau_1\ +& P(\tau_3)= \int_{0}^{\tau_3}w(\tau_3-\tau_2)d\tau_2\ +& ... \ +& P(\tau_s)= \int_{0}^{\tau_s}w(\tau_s-\tau_{s-1})d\tau_{s-1}\

Just FYI, in TeX you can do, for example, \int_0^\infty, i.e. you don't need braces in superscripts and subscripts if there's only one symbol.

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[Edlingeer]

I made some changes as you asked. Is it okay now ? sorry for the time it took me to get to this, I had some unexpected last minutes things that came up.

[DanielSank]

Woah, I'm sorry I forgot about this. I'll try to look at it this weekend.