Poisson distribution

• Is the Sample Mean of a Poisson Distribution Normal?
  For iid Poisson observations, the sample mean is a scaled Poisson variable, not exactly normal; the CLT gives its large-sample normal approximation.
• How to Compute the MGF of a Compound Poisson Distribution
  Compute a compound Poisson MGF by conditioning on the Poisson count: the result is exp(lambda times (the jump MGF minus one)).
• Expected Value of 1/(X + 1) for a Poisson Distribution
  For X ~ Poisson(λ), E[1/(X+1)] = (1−e^−λ)/λ. See the series derivation, a worked example, and the special case λ = 0.
• What Is the Expected Value of a Poisson Distribution?
  For a Poisson random variable with parameter λ, the expected value is λ. See the derivation, a worked rate example, and common mistakes.
• How to Generate a Poisson Random Variable from a Uniform One
  Use inverse-transform sampling: compare a standard uniform draw with cumulative Poisson probabilities, updating each probability recursively until you find the count.
• How to Find the Inverse of a Poisson Distribution Function
  The inverse Poisson CDF is a discrete quantile: find the smallest integer k whose cumulative probability is at least p; it is computed numerically.
• How Does a Poisson Distribution Converge to Normal?
  As a Poisson mean λ grows, centering by λ and scaling by √λ gives a standard normal limit; use a continuity correction for count probabilities.