The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find, in terms of \(n\) and \(p\), an expression for \(\operatorname { Var } ( X )\).
The mode, \(m\), of \(X\) is such that
$$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
Use the first inequality to show that
$$m \leqslant ( n + 1 ) p$$
Given that the second inequality results in
$$m \geqslant ( n + 1 ) p - 1$$
deduce that the distribution \(\mathrm { B } ( 10,0.65 )\) has one mode, and find the two values for the mode of the distribution \(B ( 35,0.5 )\).
The random variable \(Y\) has a binomial distribution with parameters 4000 and 0.00095 .
Use a distributional approximation to estimate \(\mathrm { P } ( Y \leqslant k )\), where \(k\) denotes the mode of \(Y\).
(3 marks)