| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 4 |
| 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 probabilities and possible values of X, then differentiation to find E(X). While PGFs are an advanced topic, the techniques are relatively standard once the theory is known, placing it moderately above average difficulty. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks |
|---|---|
| \(E(X) = G_X'(1) = a(4t^3 + 6t + 0 - 2t^{-3}) | _{t=1}\) — Differentiate and evaluate at \(t = 1\) OR by symmetry [M1] |
**Question 1(a):**
$G_X(t) = at\left(t + \frac{1}{t}\right)^3 = at(t^3 + 3t + 3t^{-1} + t^{-3}) = a(t^4 + 3t^2 + 3t^0 + t^{-2})$ [M1]
$X$ takes the values $4, 2, 0, -2$ [A1]
$G_X(1) = 1$ *or* sum of coefficients $= 1$ [M1]
$\Rightarrow a = \frac{1}{8}$ [A1]
**Total: 4 marks**
**Question 1(b):**
$E(X) = G_X'(1) = a(4t^3 + 6t + 0 - 2t^{-3})|_{t=1}$ — Differentiate and evaluate at $t = 1$ **OR** by symmetry [M1]
$= 1$ [**ft** $8a$] [A1, A1ft]
**Total: 2 marks**1 The discrete random variable X has probability generating function $\mathrm { G } _ { X } ( t )$ given by
$$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 }$$
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in either order, the value of $a$ and the set of values that $X$ can take.
\item Find the value of $\mathrm { E } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2020 Q1 [4]}}