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Is there a probability distribution similar to Poisson, but with controlled variance?

Poisson distribution with a lambda has a fixed mean and fixed variance; both mean and variance are equal to the $\lambda$ parameter. I am looking for a distribution to generate samples that has properties similar to Poisson, but with smaller variance instead of a variance equal to $\lambda$.

Does Poisson have a family of distributions in which mean and variance can be controlled by two different parameters?

Tom
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  • Perhaps negative binomial is what you want. – Dave Mar 24 '20 at 02:57
  • That has larger variance, not smaller as OP requests in the post – Glen_b Mar 24 '20 at 04:21
  • See Joe Hilbe's response at https://stats.stackexchange.com/questions/67385/what-is-the-appropriate-model-for-underdispersed-count-data If it pertains to count data, Hilbe is the guru. – Mike Anderson Mar 24 '20 at 03:19
  • An ordinary binomial has a variance less than the mean with $p=1-\frac{\sigma^2}{\mu}$ and $n=\frac{\mu^2}{\mu-\sigma^2}$ with the minor issue that $n$ might then not be an integer – Henry Jun 19 '23 at 21:48

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