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.

5 questions · Standard +0.2

2.04d Normal approximation to binomial5.02i Poisson distribution: random events model
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CAIE S2 2017 June Q5
9 marks Standard +0.3
5
  1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 42 )\).
    1. Use an appropriate approximating distribution to find \(\mathrm { P } ( X \geqslant 40 )\).
    2. Justify your use of the approximating distribution.
    3. 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
6 marks Moderate -0.3
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
8 marks Standard +0.3
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
10 marks Standard +0.3
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 \(\mathrm { P } ( E \leqslant 194 )\).
Pre-U Pre-U 9795/2 2016 Specimen Q2
Standard +0.3
2
  1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
  2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).