| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Topic | Probability Generating Functions |
| Type | Determine constant in PGF |
| Difficulty | Standard +0.8 This is a Further Maths probability generating function question requiring understanding that G_X(1)=1 to find the constant, expansion to identify possible values of X, and differentiation to find E(X). While PGFs are an advanced topic, the algebraic manipulations are straightforward once the method is known, making this moderately challenging but not exceptional for Further Maths. |
| Spec | 5.02a Discrete probability distributions: general |
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(The mark scheme shown is for a different Question 1 involving moment generating function $M_X(t) = \frac{k}{k-t}$, not the PGF question listed here.)1 The discrete random variable $X$ has probability generating function $\mathrm { G } _ { X } ( t )$ given by
$$\mathrm { G } _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$
where $a$ is a constant.\\
(i) Find, in either order, the value of $a$ and the set of values that $X$ can take.\\
(ii) Find the value of $\mathrm { E } ( X )$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q1}}