Explain E(X) and Var(X) relationship

A question is this type if and only if it asks to explain why p must be small for Poisson approximation using the relationship between E(X) and Var(X) for binomial distributions.

1 questions

CAIE S2 2023 June Q2
2
  1. The random variable \(W\) has a Poisson distribution.
    State the relationship between \(\mathrm { E } ( W )\) and \(\operatorname { Var } ( W )\).
  2. 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.
  3. Given that \(Y\) has the distribution \(\mathrm { B } ( 20000,0.00007 )\), use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( Y > 2 )\).