Probability Generating Functions (Edexcel A Level Further Maths: Further Statistics 1): Exam Questions

The random variable $X$ has probability generating function ${\text{G}}_X\left(t\right)$, where

${\text{G}}_X\left(t\right)=(9-8t{)}^{-\frac{1}{2}}$

Use calculus to find $\text{Var}\left(X\right)$. Show your working clearly.

Find the exact value of $\text{P}(X\le 2)$.

The independent random variables $X_1$, $X_2$, $X_3$ and $X_4$ each have the same distribution as $X$. The random variable $Y$ is defined by

$Y=X_1+X_2+X_3+X_4+2$

By finding the probability generating function of $Y$, state the name of the distribution of $Y$.

Hence, or otherwise, find $\text{P}(X_1+X_2+X_3+X_4=2)$.