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).

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The random variable $X$ has probability generating function $\mathrm { G } _ { X } ( t )$ where

Use calculus to find $\operatorname { Var } ( X )$ Show your working clearly.
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 $Y = X _ { 1 } + X _ { 2 } + 1$
By finding the probability generating function of $Y$, state the name of the distribution of $Y$
Hence, or otherwise, find $\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)$

Use the probability generating function to find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
The random variable $Y$ is defined by $Y = \frac { 1 } { 2 } ( X + 4 )$. By finding the probability distribution of $X$, or otherwise, show that $Y \sim \mathrm {~B} ( n , p )$, stating the values of $n$ and $p$.