| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 10 |
| Topic | Probability Generating Functions |
| Type | Moment generating function problems |
| Difficulty | Standard +0.3 This is a straightforward probability distribution question requiring standard techniques: finding k using ∑P(X=x)=1, computing the MGF by definition M_X(t)=E(e^{tX}), then differentiating to find mean and variance. While it involves multiple steps and the MGF concept (Further Maths content), each step follows directly from definitions with no problem-solving insight required. The calculations are routine for Further Maths students. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 1 | 2 |
| \(P(X=x)\) | 0.1 | 0.2 |
| \(M(t) = 0.1e^t + 0.2e^{2t} + 0.3e^{3t} + 0.4e^{4t}\) | M1, A1, A1 | 3 |
| Answer | Marks |
|---|---|
| \(E(X) = M'_X(0) = 3\) | M1, M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X^2) = M''_X(0) = 10\) | M1, M1, A1 | |
| \(\text{Var}(X) = 10 - 3^2 = 1\) | A1 | 7 |
### (i)
$k = 0.1$
| $x$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| $P(X=x)$ | 0.1 | 0.2 | 0.3 | 0.4 |
$M(t) = 0.1e^t + 0.2e^{2t} + 0.3e^{3t} + 0.4e^{4t}$ | M1, A1, A1 | **3**
### (ii)
$M'_X(t) = 0.1e^t + 0.4e^{2t} + 0.9e^{3t} + 1.6e^{4t}$
$E(X) = M'_X(0) = 3$ | M1, M1, A1 |
$M''_X(t) = 0.1e^t + 0.8e^{2t} + 2.7e^{3t} + 6.4e^{4t}$
$E(X^2) = M''_X(0) = 10$ | M1, M1, A1 |
$\text{Var}(X) = 10 - 3^2 = 1$ | A1 | **7**A biased tetrahedral die has faces numbered $1$ to $4$. The random variable $X$ is the number on the face of the die which is in contact with the table after the die has been thrown. It is known, for this die, that $\text{P}(X = x) = kx$ where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Determine the value of $k$ and state the moment generating function of $X$. [3]
\item Hence find $\text{E}(X)$ and $\text{Var}(X)$. [7]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q10 [10]}}