4 The moment generating function of a continuous random variable \(Y\), which has a \(\chi ^ { 2 }\) distribution with \(n\) degrees of freedom, is \(( 1 - 2 t ) ^ { - \frac { 1 } { 2 } n }\), where \(0 \leqslant t < \frac { 1 } { 2 }\).
- Find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
For the case \(n = 1\), the sum of 60 independent observations of \(Y\) is denoted by \(S\).
- Write down the moment generating function of \(S\) and hence identify the distribution of \(S\).
- Use a normal approximation to estimate \(\mathrm { P } ( S \geqslant 70 )\).