Multiple approximations in one question

A question is this type if and only if it requires using different approximations (e.g., both Poisson to normal and binomial to Poisson) in different parts of the same question.

4 questions

CAIE S2 2017 June Q5
5
  1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 42 )\).
    (a) Use an appropriate approximating distribution to find \(\mathrm { P } ( X \geqslant 40 )\).
    (b) Justify your use of the approximating distribution.
  2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 60,0.02 )\).
    (a) Use an appropriate approximating distribution to find \(\mathrm { P } ( Y > 2 )\).
    (b) Justify your use of the approximating distribution.
CAIE S2 2023 November Q1
1
  1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 25 )\).
    Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 30 )\).
  2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 100 , p )\) where \(p < 0.05\). Use the Poisson approximation to the binomial distribution to write down an expression, in terms of \(p\), for \(\mathrm { P } ( Y < 3 )\).
OCR S2 2005 June Q3
3
  1. The random variable \(X\) has a \(\mathrm { B } ( 60,0.02 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( X \leqslant 2 )\).
  2. The random variable \(Y\) has a \(\operatorname { Po } ( 30 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( Y \leqslant 38 )\).
OCR S2 2010 June Q6
6
  1. The random variable \(D\) has the distribution \(\operatorname { Po } ( 24 )\). Use a suitable approximation to find \(P ( D > 30 )\).
  2. An experiment consists of 200 trials. For each trial, the probability that the result is a success is 0.98 , independent of all other trials. The total number of successes is denoted by \(E\).
    1. Explain why the distribution of \(E\) cannot be well approximated by a Poisson distribution.
    2. By considering the number of failures, use an appropriate Poisson approximation to find \(\mathrm { P } ( E \leqslant 194 )\).