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.

2 questions

CAIE S2 2012 June Q5
5 A random variable \(X\) has the distribution \(\operatorname { Po } ( 3.2 )\).
  1. A random value of \(X\) is found.
    (a) Find \(\mathrm { P } ( X \geqslant 3 )\).
    (b) Find the probability that \(X = 3\) given that \(X \geqslant 3\).
  2. Random samples of 120 values of \(X\) are taken.
    (a) Describe fully the distribution of the sample mean.
    (b) Find the probability that the mean of a random sample of size 120 is less than 3.3.
CAIE S2 2018 November Q7
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.
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