- The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$
- 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 \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)