[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Find the generating functions, both ordinary [math]h(z)[/math]

and moment [math]g(t)[/math], for the following discrete probability distributions.

• The distribution describing a fair coin.
• The distribution describing a fair die.
• The distribution describing a die that always comes up 3.
• The uniform distribution on the set [math]\{n,n+1,n+2,\ldots,n+k\}[/math].
• The binomial distribution on [math]\{n,n+1,n+2,\ldots,n+k\}[/math].
• The geometric distribution on [math]\{0,1,2,\ldots,\}[/math] with [math]p(j) = 2/3^{j + 1}[/math].