Question 7 - A-Level Maths

Exam Board	AQA
Module	S3 (Statistics 3)
Year	2011
Session	June
Topic	Poisson Distribution
Type	Proving Poisson properties from first principles

The random variable \(X\) has a Poisson distribution with \(\mathrm { E } ( X ) = \lambda\).
1. Prove, from first principles, that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\).
2. Hence deduce that \(\operatorname { Var } ( X ) = \mathrm { E } ( X )\).
The random variable \(Y\) has a Poisson distribution with \(\mathrm { E } ( Y ) = 2.5\). Given that \(Z = 4 Y + 30\) :
1. show that \(\operatorname { Var } ( Z ) = \mathrm { E } ( Z )\);
2. give a reason why the distribution of \(Z\) is not Poisson.
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