The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
Prove that \(\mathrm { E } ( X ) = n p\).
Given that \(\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }\), show that \(\operatorname { Var } ( X ) = n p ( 1 - p )\).
Given that \(X\) is found to have a mean of 3 and a variance of 2.97, find values for \(n\) and \(p\).
Hence use a distributional approximation to estimate \(\mathrm { P } ( X > 2 )\).
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.