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I have following equation,

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.

hardmath
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Nikki
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    It looks very messy, may I ask why you think P can be solved ? – AgentS Feb 27 '18 at 21:22
  • @rsadhvika Because it has specific roots for P, I don't know how to solve it but it has been solved before. – Nikki Feb 27 '18 at 21:23
  • Looks like there are more than one P... – Samuel Feb 27 '18 at 21:23
  • I thought the P stood for permutation much like how C stands for combination, but anyway... – Mr Pie Feb 27 '18 at 21:26
  • @Samuel other P which is Pj is somehow related to Pi, for i = 1, that summation is going to be zero and from that we could have values for i with higher numbers. – Nikki Feb 27 '18 at 21:27
  • @user477343 no P is actually the probability of having non zero events in one channel, for example channel i. – Nikki Feb 27 '18 at 21:28
  • @N.Sh: Can you give an example with a specific function $S(n)$, and a specific sequence $(N_i)$? – quasi Feb 27 '18 at 21:30
  • @N.Sh: Also, what is the definition of the variable $D$? – quasi Feb 27 '18 at 21:32
  • @quasi S(n) is basically the distribution for n, let's say n could be an integer between 1 and 6 and S(n) is giving the probability of having that specific n, it could be something like this : [0.1, 0.0, 0.3, 0.4, 0.1, 0.1], Ni is the number of counts in each channel, if we have 5 channels, it could be something like: [10, 15, 0, 23, 0], D is basically and integer that is known, channels with D distance from selected channel could influence the probability of that channel. – Nikki Feb 27 '18 at 21:33
  • @N.Sh: $P_i$ is the probability of what? – quasi Feb 27 '18 at 21:36
  • @quasi we have like 5 channels, each channel could have a probability that we could detect an event in them. That is P for each channel. – Nikki Feb 27 '18 at 21:38
  • @quasi maybe the better way to explain P is to say that it is the probability of having r events of n events happen in channel i – Nikki Feb 27 '18 at 21:45

1 Answers1

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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 . . .

enter image description here

quasi
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