3 The discrete random variable \(X\) has probability generating function \(\frac { t } { a - b t }\), where \(a\) and \(b\) are constants.
- Find a relationship between \(a\) and \(b\).
- Use the probability generating function to find \(\mathrm { E } ( X )\) in terms of \(a\), giving your answer as simply as possible.
- Expand the probability generating function as a power series, as far as the term in \(t ^ { 3 }\), giving the coefficients in terms of \(a\) and \(b\).
- Name the distribution for which \(\frac { t } { a - b t }\) is the probability generating function, and state its parameter(s) in terms of \(a\).