Sample mean distribution of Poisson

A question is this type if and only if it asks about the distribution of the sample mean from multiple independent Poisson observations, typically using CLT for large samples.

4 questions · Standard +0.4

5.02n Sum of Poisson variables: is Poisson5.05a Sample mean distribution: central limit theorem
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CAIE S2 2024 June Q7
13 marks Standard +0.3
7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 1.9 )\) and \(\operatorname { Po } ( 2.2 )\) respectively.
  1. Find \(\mathrm { P } ( X + Y < 4 )\). \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_74_1581_406_322} \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_75_1581_497_322}
  2. Find the probability that \(X = 2\) given that \(X + Y < 4\). \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_2715_35_144_2012}
  3. A sample of 60 randomly chosen pairs of values of \(X\) and \(Y\) is taken,and the value of \(X + Y\) is calculated for each pair.The sample mean of these 60 values is found. Find the probability that the sample mean of \(X + Y\) is less than 4.0 .
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2018 November Q7
11 marks Standard +0.8
7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively.
  1. Find \(\mathrm { P } ( X + Y = 3 )\).
  2. Given that \(X + Y = 3\), find \(\mathrm { P } ( X = 2 )\).
  3. A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is more than 2.2.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2016 June Q6
9 marks Standard +0.3
\(X\) and \(Y\) are independent random variables with distributions \(\mathrm{Po}(1.6)\) and \(\mathrm{Po}(2.3)\) respectively.
  1. Find \(\mathrm{P}(X + Y = 4)\). [3]
A random sample of 75 values of \(X\) is taken.
  1. State the approximate distribution of the sample mean, \(\overline{X}\), including the values of the parameters. [2]
  2. Hence find the probability that the sample mean is more than 1.7. [3]
  3. Explain whether the Central Limit theorem was needed to answer part (ii). [1]
CAIE S2 2002 November Q5
8 marks Standard +0.3
\(X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.
  1. Find P\((X + Y > 3)\). [4]
  2. A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4. [4]