6 The discrete random variable \(X\) takes the values 0 and 1 with \(\mathrm { P } ( X = 0 ) = q\) and \(\mathrm { P } ( X = 1 ) = p\), where \(p + q = 1\).
- Write down the probability generating function of \(X\).
The sum of \(n\) independent observations of \(X\) is denoted by \(S\).
- Write down the probability generating function of \(S\), and name the distribution of \(S\).
- Use the probability generating function of \(S\) to find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
- The independent random variables \(Y\) and \(Z\) are such that \(Y\) has the distribution \(\mathrm { B } \left( 10 , \frac { 1 } { 2 } \right)\), and \(Z\) has probability generating function \(\mathrm { e } ^ { - ( 1 - t ) }\). Find the probability that the sum of one random observation of \(Y\) and one random observation of \(Z\) is equal to 2 .