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Vamos a $p_{-}$ $p_{+}$ la probabilidad de una bacteria muere o se divide en dos bacterias, respectivamente. La probabilidad de una bacteria que genera $n$ bacterias se
$$P_{n} = p_{-}\delta_{n,0}\ +\ p_{+}\delta_{n,2}\tag{1}$$
Dada una población de $N$ bacterias vamos a calcular la probabilidad de
${\cal P}_{N \to N'}$ de la población será de $N'$. ${\cal P}_{N \to N'}$ está dada por:
\begin{align}
{\cal P}_{N \to N'}&=\sum_{n_{1}=0}^{\infty}P_{n_{1}}\ldots
\sum_{n_{N}=0}^{\infty}P_{n_{N}}\,\delta_{n_{1} + \cdots + n_{N},N'}
\\[3mm]&=\sum_{n_{1}=0}^{\infty}P_{n_{1}}\ldots\sum_{n_{N}=0}^{\infty}P_{n_{N}}\
\overbrace{\int_{\verts{z} = 1}{1 \over z^{-n_{1} - \cdots - n_{N} + N' + 1}}
\,{\dd z \over 2\pi\ic}}^{\ds{\delta_{n_{1} + \cdots + n_{N},N'}}}
\\[3mm]&=\int_{\verts{z} =1}\pars{\sum_{n = 0}^{\infty}P_{n}z^{n}}^{N}\,
{1 \over z^{N' + 1}}\,{\dd z \over 2\pi\ic}
=\int_{\verts{z} =1}\pars{p_{-} + p_{+}z^{2}}^{N}\,{1 \over z^{N' + 1}}
\,{\dd z \over 2\pi\ic}
\end{align}
donde hemos utilizado la expresión $\pars{1}$.
\begin{align}
{\cal P}_{N \to N'}&=
\int_{\verts{z} = 1}\sum_{\ell = 0}^{N}
{N \choose \ell}p_{-}^{N - \ell}\pars{p_{+}z^{2}}^{\ell}\,{1 \over z^{N' + 1}}
\,{\dd z \over 2\pi\ic}
\\[3mm]&=\sum_{\ell = 0}^{N}{N \choose \ell}p_{-}^{N -\ell}p_{+}^{\ell}\
\overbrace{\int_{\verts{z} = 1}{1 \over z^{N' + 1 - 2\ell}}\,{\dd z \over 2\pi\ic}}
^{\ds{\delta_{2\ell,N'}}}\tag{2}
\end{align}
A partir de la expresión de $\pars{2}$ llegamos a la conclusión de:
$$
{\cal P}_{N \N'}
=\left\lbrace%
\begin{array}{lcl}
{N \choose N'/2}p_{-}^{N - N'/2}\ p_{+}^{N'/2}
& \mbox{if} & N'\ \mbox{is even and}\ 0 \leq N' \leq 2N
\\[2mm]
0, &&\mbox{otherwise}
\end{array}\right.
$$
Para el presente pregunta
$$
p_{-}^{N - N'/2}\ p_{+}^{N'/2}
=
\pars{1 \over 3}^{N - N'/2}\pars{2 \más de 3}^{N'/2} = {2^{N'/2} \más de 3^{N}}
$$
$$\color{#00f}{\large%
{\cal P}_{N \N'}
=\left\lbrace%
\begin{array}{lcl}
{1 \over 3^{N}}\,{N \choose N'/2}2^{N'/2}
& \mbox{if} & N'\ \mbox{is even and}\ 0 \leq N' \leq 2N
\\[2mm]
0, &&\mbox{otherwise}
\end{array}\right.}
$$
Con esta expresión para ${\cal P}_{N \to N'}$ podemos responder a muchas preguntas acerca de la población de bacterias.
