# What are the mean and standard deviation of a probability density function given by Pr(X=k) = (4^ke^-4)/(k!)  for k in {0,1,2... oo}?

$\mu = 4$, $\sigma = 2$.
The probability density function given by P(X=k)=(4^ke^-4)/(k!) is the Poisson distribution with 4 degrees of freedom. This is because the PDF of the Poisson distribution with $\lambda$ degrees of freedom is P(X=k)=(lambda^ke^-lambda)/(k!). The mean and variance of this distribution are both $\lambda$ (proofs can be found at https://proofwiki.org/wiki/Expectation_of_Poisson_Distribution and https://proofwiki.org/wiki/Variance_of_Poisson_Distribution). Since variance is equal to ${\sigma}^{2}$, we know that $\sigma = 2$.