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 $$ \begin{aligned} \mathcal{G}_{X}(z) &= \sum_{k = 1}^{\infty} z^k\frac{\lambda^k e^{-\lambda}}{k!} \\ &= e^{-\lambda}\sum_{k = 1}^{\infty} \frac{(\lambda z)^{k}}{k!} \\ &= e^{-\lambda}e^{\lambda z} \\ &= e^{\lambda(z - 1)} \end{aligned} $$

Honours Thesis This short post contains links to my honours thesis and honours presentation. The topic of my thesis was to create a computational structure for a semi-parametric vector generalized linear model. This was based on the earlier work of my supervisor, who had built up the theoretical framework for this model over a series of papers with other collaborators. Papers https://www.tandfonline.com/doi/abs/10.1080/01621459.2013.824892 R Package for Univariate Case Univariate model R package CRAN Link Relevant Links The following links contain some of my work on this topic.

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