4 The random variable \(U\) has the distribution \(\operatorname { Geo } ( p )\).
- Show, from the definition, that the probability generating function ( pgf ) of \(U\) is given by
$$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$
where \(q = 1 - p\).
- Explain why the condition \(| t | < \frac { 1 } { q }\) is necessary.
- Use the pgf to obtain \(\mathrm { E } ( U )\).
Each packet of Corn Crisp cereal contains a voucher and \(20 \%\) of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let \(X\) denote the number of packets bought by a family up to and including the one from which the \(4 ^ { \text {th } }\) gold star is obtained.
- Obtain the pgf of \(X\).
- Find \(\mathrm { P } ( X = 6 )\).