how to rewrite this equation to solve for P?

Question

I have following equation,

How could I rewrite this equation that I could solve it for P?

Ni is number of counts in i, N is the total number of counts, S(n) is a known function, the C part is binomial coefficient and P is actually the probability.

Answer 1

Changing a few letters for notational convenience, you have the system $$ w_i = W \sum_{n=1}^m s(n) \left( \sum_{r=1}^n {\small{\binom{n}{r}}}p_i^r\left(1-p_i-\sum_{j=i-d-1}^{i-1}p_j\right)^{n-r} \right) $$ of $m$ equations, where

• $w_1,...,w_m$ are given nonnegative integers, representing channel counts.$\\[4pt]$
• $W= w_1 + \cdots + w_m$.$\\[4pt]$
• $s(1),...,s(m)$ are given probabilities, with $s(1) + \cdots + s(m)=1$.$\\[4pt]$
• $d$ is a given nonnegative integer.$\\[4pt]$
• $p_1,...,p_m$ are unknown probabilities.

Thus, you have a system of $m$ equations, in $m$ unknowns.

Let's assume that the real-world interpretation ensures that a solution $(p_1,...,p_m)$ exists, and is unique.

The bad news is that the equations are non-linear.

The good news is that the $i$-th equation depends only on a subset of $\{p_1,...,p_i\}$, so you can solve the system by solving the equations successively, one at a time, starting with $i=1$.

I doubt that you can get a closed form for $p_i$, but solving numerically shouldn't be a problem.

Here's a Maple program which accomplishes the task . . .