The following is a somewhat mathematical question, but I am interested in using the idea to define a set of quantum measurement operators defined as described in the answer to this post.(adsbygoogle = window.adsbygoogle || []).push({});

Question:

The Poisson Distribution##Pr(M|\lambda)## is given by $$Pr(M|\lambda) = \frac{e^{-\lambda}\lambda^M}{M!}~~~~M = 0,1,2...$$ with mean ##\lambda##. In addition, the distribution has the property that $$\int_{0}^{\infty} Pr(M|\lambda) d \lambda = 1~~~~~~~~~~~~~~(*)$$ as is proved in the post. I am interested in adapting this Poisson Distribution so that it can include an additional parameter (say ##\mu##) which determines the width of the distribution (much like thevariancedoes for the normal distribution. Does anyone have any ideas of how this can be achieved while still maintaining that property ##(*)## holds?

Note I want to specifically work with the poisson distribution rather than the normal distribution due to how the shape of the poisson distribution at zero.

Thanks for your time and let me know if you have any queries.

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# A Quantum measurement operators with Poisson distribution

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