Use PGF to find mean and variance

Given a PGF (as polynomial or function), use differentiation (G'(1) for mean, G''(1) + G'(1) - [G'(1)]² for variance) to calculate E(X) and Var(X).

1 questions

Edexcel FS1 2024 June Q6
  1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$
  1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
  2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
  3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
  4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)