Poisson Distribution The general form of the pdf for the Poisson distribution is $$ f(x; \lambda) = \mathbb{P}(X = x) = \frac{\lambda^x e^{-\lambda}}{x!} $$ And in this instance we say that the R.V \(X \sim P(\lambda)\). PGF The Probability Generating function of a Poisson Distribution has the following form $$ \begin{aligned} \mathcal{G}_{X}(z) &= \sum_{k = 1}^{\infty} z^k\frac{\lambda^k e^{-\lambda}}{k!} \\ &= e^{-\lambda}\sum_{k = 1}^{\infty} \frac{(\lambda z)^{k}}{k!} \\ &= e^{-\lambda}e^{\lambda z} \\ &= e^{\lambda(z - 1)} \end{aligned} $$

The Normal Distribution The normal or Gaussian distribution is perhaps the most common distribution occuring in nature. This is due to the relatively special properties it has for larger sample sizes. However, these properties will be the topic of another blog post. Instead, in this post we are simply going to outline the basic mathematical structure of the normal distribution. Probability Density Function The probability density function of a $$ \mathcal{N}(\mu, \sigma^2) $$ normal distribution is