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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?

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  • $\begingroup$ Perhaps negative binomial is what you want. $\endgroup$
    – Dave
    Mar 24, 2020 at 2:57
  • $\begingroup$ That has larger variance, not smaller as OP requests in the post $\endgroup$
    – Glen_b
    Mar 24, 2020 at 4:21

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See Joe Hilbe's response at What is the appropriate model for underdispersed count data?

If it pertains to count data, Hilbe is the guru.

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