The variance of $X \sim$Poisson$(\lambda)$ is $\lambda$, which is a constant. Intuitively, I take this to be that the values of $X$ are symmetrically distributed; however, many Poisson distributions I have seen are wildly skewed to one side. So how can I reconcile these 2 things? Why is the Poisson distribution not necessarily symmetrical even though it has constant variance?
2026-04-01 01:02:38.1775005358
Why is the Poisson distribution not necessarily symmetrical even though it has constant variance?
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Actually, the variance of a distribution is always a function of the parameters of that distribution. If you take the parameters ($\lambda$ in this case) as constants, then the variance is also always a constant.
This has nothing to do with symmetry of the distribution, whioch is better expressed by the Skewness of the distribution.