Find probabilities from PGF

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

1 questions

OCR S4 2017 June Q7
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 \(\mathrm { E } ( Y )\).