The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
Given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find an expression for \(\operatorname { Var } ( X )\).
The random variable \(Y\) has a binomial distribution with \(\mathrm { E } ( Y ) = 3\) and \(\operatorname { Var } ( Y ) = 2.985\).
Find values for \(n\) and \(p\).
The random variable \(U\) has \(\mathrm { E } ( U ) = 5\) and \(\operatorname { Var } ( U ) = 6.25\).
Show that \(U\) does not have a binomial distribution.
The random variable \(V\) has the distribution \(\operatorname { Po } ( 5 )\) and \(W = 2 V + 10\).
Show that \(\mathrm { E } ( W ) = \operatorname { Var } ( W )\) but that \(W\) does not have a Poisson distribution.
The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected.
Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative. [0pt]
[3 marks]