| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 13 |
| Topic | Probability Generating Functions |
| Type | Find PGF from probability distribution |
| Difficulty | Standard +0.8 This is a Further Maths probability generating functions question requiring understanding of PGF definition, properties under summation of independent variables, and differentiation to find expectation and coefficient extraction. While PGFs are advanced, the actual calculations are straightforward once the method is known, making this moderately challenging but not exceptional for Further Maths students. |
| Spec | 5.02a Discrete probability distributions: general |
| \(x\) | - 1 | 2 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 3 }\) | \(\frac { 2 } { 3 }\) |
Total: 11 marks
**Part (i):**
$\frac{1}{3}t^{-1} + \frac{2}{3}t^2$ **B2** (B1 for each; if $x$ used in an otherwise correct expression then SC1. Condone further consistent use in part (ii).)
**Part (ii)(a):**
$G_{10}(t) = (\frac{1}{3}t^{-1} + \frac{2}{3}t^2)^{10}$ oe **M1** $([G(t)]^{10})$
Differentiate (could be any wrong G) and put $t = 1$ **M1**
$G'_{10}(t) = 10(-\frac{1}{3}t^{-2} + \frac{4}{3}t)(\frac{1}{3}t^{-1} + \frac{2}{3}t^2)^9$ **A1** Correct derivative in any form
$G'(1) = 10$ **A1** Answer 10 only. www. SC: $E(X) = 1$ B1, $E(T) = 10$ "$E(X)$" B1ft. Max 2/4
**Part (ii)(b):**
Attempt coefficient of $t^8$ **M1**
Coefficient of $t^8 = {}^{10}C_4 \left(\frac{1}{3}\right)^4\left(\frac{2}{3}\right)^6$ **A1** Correct expression (term or coefficient)
$= 4480/19683$ or awrt $0.228$ **A1** SC: 6 2s and 4 –1s: ${}^{10}C_4\left(\frac{1}{3}\right)^4\left(\frac{2}{3}\right)^6$ M1, 0.228 A1 (Max 2/3)
Total: **11 marks**2 A discrete random variable $X$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
$x$ & - 1 & 2 \\
\hline
$\mathrm { P } ( X = x )$ & $\frac { 1 } { 3 }$ & $\frac { 2 } { 3 }$ \\
\hline
\end{tabular}
\end{center}
(i) Write down the probability generating function of $X$.\\
(ii) $T$ is the sum of ten independent observations of $X$. Use the probability generating function of $T$ to find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { E } ( T )$,
\item $\mathrm { P } ( T = 8 )$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q2 [13]}}