5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
- Given that
$$\mathrm { E } ( X ) = n p \quad \text { and } \quad \mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$$
find an expression for \(\operatorname { Var } ( X )\).
- Given that \(X\) has a mean of 36 and a standard deviation of 4.8:
- find values for \(n\) and \(p\);
- use a distributional approximation to estimate \(\mathrm { P } ( 30 < X < 40 )\).