Derive binomial mean and variance

Prove or use algebraic manipulation to derive expressions for E(X) and Var(X) for binomial distributions.

2 questions

AQA S3 2009 June Q5
5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
  1. 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 \(\operatorname { Var } ( X )\).
  2. Given that \(X\) has a mean of 36 and a standard deviation of 4.8:
    1. find values for \(n\) and \(p\);
    2. use a distributional approximation to estimate \(\mathrm { P } ( 30 < X < 40 )\).
AQA S3 2007 June Q6
6
  1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
    1. Prove that \(\mathrm { E } ( X ) = n p\).
    2. Given that \(\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }\), show that \(\operatorname { Var } ( X ) = n p ( 1 - p )\).
    3. Given that \(X\) is found to have a mean of 3 and a variance of 2.97, find values for \(n\) and \(p\).
    4. Hence use a distributional approximation to estimate \(\mathrm { P } ( X > 2 )\).
  2. Dressher is a nationwide chain of stores selling women's clothes. It claims that the probability that a customer who buys clothes from its stores uses a Dressher store card is 0.45 . Assuming this claim to be correct, use a distributional approximation to estimate the probability that, in a random sample of 500 customers who buy clothes from Dressher stores, at least half of them use a Dressher store card.