Sum of random variables dependent on an ODE Let us assume we divide a piece of membrane into $n$ parts. Every part contributes with a specific electric potential, which is represented by $X_i\ $ $(i=1,\dots,n)$ random variables. So with probability $p_i$ the part $X_i$ is sending an impulse (mostly this impulse should be in the range of $[0,1]$). 


*

*All $X_i$ are independent - maybe we can assume a $N(0,1)$ distribution. 

*The $X_i$ are dependent on the total membrane voltage $V(t)$. 

I want to show that the standard deviation of the sum $$\sum_{i=1}^{n}X_i$$ grows proportional to $\sqrt{n}$. 

I can show that the standard deviation grows proportional to $\sqrt{n}$, when neglecting the dependency on $V(t)$.
But I don't know how to deal with the dependency on a function. Maybe it would be a good idea to model the dependency regarding $dV/dt$?
Shall I look at $\sum_{i=1}^{n}V(X_i)$?
I don't know how to handle this issue.

Suppose that when the voltage is $V$, the standard deviation of $X_i$ is $\sigma(V)$. Then the standard deviation is
$$
\operatorname{St. Dev}(\textstyle\sum_{i=1}^n X_i)=\sigma(V(t))\sqrt{n}.
$$
Without more knowledge of the dependence of $\sigma$ on $V$, this is all you can say. 
