Timeline for Solving recursion / finding generating function of a probability mass function
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Jan 22, 2017 at 20:26 | vote | accept | Matjaž Krnc | ||
| Mar 18, 2016 at 11:37 | comment | added | Matjaž Krnc | @esg : Sure, I actually found some great literature that is helping me a lot with this problem. The particular version of the problem is slightly different (this here is a simplified version), since reproduction function changes with each generation. Still, this approach seems to work to some extent. Thanks again! | |
| Mar 9, 2016 at 20:01 | comment | added | esg | Please note the obvious typo, should be ${Z_n \over m^n}$ instead of ${Z_n \over m}$ above. | |
| Mar 5, 2016 at 22:39 | vote | accept | Matjaž Krnc | ||
| Jan 22, 2017 at 20:26 | |||||
| Mar 5, 2016 at 22:39 | comment | added | Matjaž Krnc | Thanks guys! The connection of my problem with Galton-Watson process, as well as with Mandelbrot fractal is most appreciated! I will make sure to study G-W process in detail; and close this question. | |
| Mar 4, 2016 at 18:24 | comment | added | esg | $F_n$ ``counts'' the no. $Z_n$ of individuals in the $n-$th generation of a Galton-Watson process with reproduction function $F(s)=qs+ps^2$. Thus (look up the literature) $\mathbb{E} Z_n=m^n$, $\mathrm{Var}(Z_n)=\sigma^2 m^{n-1}{m^n-1 \over m-1}$ where $m=\mathbb{E}(Z_1)=1+p, \sigma^2={\mathrm{Var}}(Z_1)=pq$. Further $${Z_n \over m}\longrightarrow W\;\;\mbox{a.s.}$$ where $W$ is positive on $\{Z_n\longrightarrow \infty\}$, and the Laplace transform $\ell$ of $W$ is characterized by the the equation $\ell(mt)=q\ell(t)+p\ell(t)^2$, and right derivative $-\ell^\prime(0)=1$ in $0$. | |
| Mar 4, 2016 at 14:01 | history | edited | Matjaž Krnc | CC BY-SA 3.0 |
the calculations of first few members of $F_n$ seems to be not relevant, and are removed. Some additional info is added.
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| Mar 4, 2016 at 13:24 | comment | added | Matjaž Krnc | As is written, I am initially looking for $f_n(k)$. The generating function approach was just one way of representing problem with in a simpler form (or so I thought). As you pointed out in your answer it looks that this is not the case. | |
| Mar 1, 2016 at 12:04 | answer | added | SM2 | timeline score: 2 | |
| Feb 29, 2016 at 23:26 | comment | added | Mark Fischler | Are you looking for a closed-form expression for the generating function, as a function of $x$ and $p$, or would a closed form expression for $f_n(k)$ be satisfactory? for | |
| Feb 29, 2016 at 16:40 | history | edited | Matjaž Krnc | CC BY-SA 3.0 |
changed title
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| Feb 29, 2016 at 16:31 | history | asked | Matjaž Krnc | CC BY-SA 3.0 |