Q.State the mean and variance of the Binomial, Poisson and Normal distributions, and explain the special relationship that holds between the mean and variance of the Poisson distribution but not, in general, for the other two.

Comparing the mean and variance formulas of all three distributions side by side reveals that only the Poisson distribution ties its two summary numbers together into a single parameter. …

Binomial: mean $= np$, variance $= npq$. Since $q = 1-p \le 1$, the variance $npq$ is always less than or equal to the mean $np$, with equality only in the degenerate case $q = 1$ (i.e. $p = 0$, no successes possible at all). In every realistic case, Binomial variance is strictly less than its mean.

Poisson: mean $= \lambda$, variance $= \lambda$. These are exactly equal, for every value of $\lambda$. This is not a coincidence of a particular example — it follows directly from the Poisson distribution having only one parameter, $\lambda$, which simultaneously fixes both its mean and its variance. This property is often used in practice as a diagnostic: if a real dataset's sample variance is very different from its sample mean, a Poisson model may not be a good fit for that data. …

Students sometimes assume the Poisson property "mean = variance" is a general fact true of all distributions, or conversely assume it can never happen for a Binomial — it can happen for Binomial only …