Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The Poisson cdf can be written as $F(k, \lambda) = e^{-\lambda} \sum\limits_{i = 0}^{k}\frac{\lambda^{i}}{i!}$. I was wondering if it was possible to solve this equation for $\lambda$? That is, I know $k$ and the value of $F$, but I want to know what value of $\lambda$ makes the equation equal $F$.


share|cite|improve this question
Shouldn't the lower limit of the sum be $i=0$? You can solve it numerically, but not (unless k=0) algebraically. – Ross Millikan Sep 28 '11 at 17:05
Do you know of any resource that can give me more information on how to go about solve it numerically? – Erich Peterson Sep 28 '11 at 17:08
Any numerical analysis book. I like Numerical Recipes,, chapter 9. Obsolete versions are free on-line – Ross Millikan Sep 28 '11 at 17:12
up vote 4 down vote accepted

$F(k,\lambda) = Q(k+1,\lambda)$ (see, for example, here), where $Q(s,x) = \Gamma(s,x)/\Gamma(s)$ is the regularized incomplete gamma function. This is a well-known enough special function that some common software packages have an implementation for its inverse in $x$. For example, Mathematica uses the command InverseGammaRegularized. Wolfram|Alpha uses the same command and is freely available, so that may be preferred.

As an illustration, suppose $F = 0.5$ and $k = 10$. Wolfram|Alpha says that InverseGammaRegularized[11,0.5] = 10.668522..., and we can verify this by calculating $$e^{-10.6685} \sum_{i=0}^{10} \frac{10.6685^i}{i!},$$ which Wolfram|Alpha says is $0.5$.

share|cite|improve this answer
Should you need to do things yourself... you can also start up with Wilson-Hilferty and polish off with Newton-Raphson. – J. M. Sep 28 '11 at 17:40

You can use Excel solver in the analysis toolpak add-in. Use the evolutionary feature. Minimize the absolute value of the difference between your desired F and the cell that calculates the CDF of your best guess at lambda. Give it about 30 seconds to run and you'll have your answer.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.