Find probabilities from PGF

Given a PGF, extract P(X = r) by finding the coefficient of t^r, either by expansion or differentiation.

2 questions · Challenging +1.0

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OCR S4 2017 June Q7
9 marks Challenging +1.2
7 The discrete random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t ) = \frac { 1 } { 126 } t \left( 64 - t ^ { 6 } \right) \left( 1 - \frac { t } { 2 } \right) ^ { - 1 }\).
  1. Find \(\mathrm { P } ( Y = 3 )\).
  2. Find \(\mathrm { E } ( Y )\).
CAIE Further Paper 4 2021 June Q4
3 marks Standard +0.8
  1. Find P\((X = 4)\). [3]