The random variable \(W\) has a Poisson distribution.
State the relationship between \(\mathrm { E } ( W )\) and \(\operatorname { Var } ( W )\).
The random variable \(X\) has the distribution \(\mathrm { B } ( n , p )\). Jyothi wishes to use a Poisson distribution as an approximate distribution for \(X\).
Use the formulae for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\) to explain why it is necessary for \(p\) to be close to 0 for this to be a reasonable approximation.
Given that \(Y\) has the distribution \(\mathrm { B } ( 20000,0.00007 )\), use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( Y > 2 )\).