Cumulative Binomial Distribution function , Solve for n (trials)

Question

how can one solve for $n$ in the Cumulative Binomial Distribution Function $P=\sum_{i=0}^{i=c-1} {n \choose i} p^{i}(1-p)^{n-i}$.

thanks in advance,

D.

Answer 1

I dont think it is possible to solve for $n$ in this form. If $n$ is large enough, this is approximated well by a Poisson Distribution.

We can think in terms of expected value by letting $k = np$, that is over the sample size $n$, if an even occurs with probability $p$, then it occurs on average, $k$ times.

Then each term can be rewritten as:

$ \dbinom{n}i \left( \frac{k}{n} \right)^i\left(1- \frac{k}n \right)^{n-i}$

$= \frac{n(n-1) \cdots (n-i+1)}{i!} \frac{k^i}{n^i} \left(1- \frac{k}n \right)^{n-i}$

$ = \frac{n(n-1) \cdots (n-i+1)}{n^i} \frac{k^i}{i!} \left(1- \frac{k}n \right)^{n-i}$

which can be approximated by

$\frac{n^i}{n^i} \frac{k^i}{i!} e^{-k} \cdot 1$

$ = \frac{k^i}{i!} e^{-k} $