Use MGF to find moments

A question is this type if and only if it asks to use the MGF (by differentiation) to find mean, variance, or higher moments of a distribution.

2 questions · Standard +0.3

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OCR S4 2015 June Q5
9 marks Standard +0.3
5 The random variable \(X\) has a Poisson distribution with mean \(\lambda\). It is given that the moment generating function of \(X\) is \(e ^ { \lambda \left( e ^ { t } - 1 \right) }\).
  1. Use the moment generating function to verify that the mean of \(X\) is \(\lambda\), and to show that the variance of \(X\) is also \(\lambda\).
  2. Five independent observations of \(X\) are added to produce a new variable \(Y\). Find the moment generating function of \(Y\), simplifying your answer.
Pre-U Pre-U 9795/2 2018 June Q3
Standard +0.3
3 The moment generating function of a random variable \(X\) is \(( 1 - 2 t ) ^ { - 3 }\).
  1. Find the mean and variance of \(X\).
  2. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Find \(\mathrm { E } \left[ \left( X _ { 1 } + X _ { 2 } \right) ^ { 3 } \right]\).