[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]

Let [math]p[/math] and [math]p'[/math] be the two distributions

[[math]] p = \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr 1/3 & 0 & 0 & 2/3 & 0 \cr}\ , [[/math]]

[[math]] p' = \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr 0 & 2/3 & 0 & 0 & 1/3 \cr}\ . [[/math]]

• Show that [math]p[/math] and [math]p'[/math] have the same first and second moments, but not the same third and fourth moments.
• Find the ordinary and moment generating functions for [math]p[/math] and [math]p'[/math].