| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Find PGF of sum of independent variables |
| Difficulty | Standard +0.8 This is a standard PGF question requiring knowledge that G_Y(t) = [G_X(t)]^2 for sum of independent variables, followed by polynomial expansion and differentiation. While mechanical, it involves Further Maths content (PGFs are UFM-only), multiple steps including algebraic manipulation of a cubic squared, and applying PGF properties for expectation/variance. More demanding than typical A-level but routine for Further Maths students who know the theory. |
| Spec | 5.02a Discrete probability distributions: general |
| Answer | Marks |
|---|---|
| \(G_X(t) = 0.2t + 0.5t^2 + 0.3t^3\) | B1 |
| \(G_Y(t) = \left(0.2t + 0.5t^2 + 0.3t^3\right)^2\) | M1 |
| \(= 0.04t^2 + 0.2t^3 + 0.37t^4 + 0.3t^5 + 0.09t^6\) | A1 |
| Answer | Marks |
|---|---|
| \(G'_Y(t) = 0.08t + 0.6t^2 + 1.48t^3 + 1.5t^4 + 0.54t^5\) | M1 |
| \(E(Y) = 4.2\) | A1 |
| \(G''_Y(t) = 0.08 + 1.2t + 4.44t^2 + 6t^3 + 2.7t^4\) | M1 |
| Use \(G''_Y(1) + G'_Y(1) - (G'_Y(1))^2\) | M1 |
| \(\text{Var}(Y) = 14.42 + 4.2 - 4.2^2 = 0.98\) | A1 |
## Question 4:
**Part 4(a):**
$G_X(t) = 0.2t + 0.5t^2 + 0.3t^3$ | B1 |
$G_Y(t) = \left(0.2t + 0.5t^2 + 0.3t^3\right)^2$ | M1 |
$= 0.04t^2 + 0.2t^3 + 0.37t^4 + 0.3t^5 + 0.09t^6$ | A1 |
**Part 4(b):**
$G'_Y(t) = 0.08t + 0.6t^2 + 1.48t^3 + 1.5t^4 + 0.54t^5$ | M1 |
$E(Y) = 4.2$ | A1 |
$G''_Y(t) = 0.08 + 1.2t + 4.44t^2 + 6t^3 + 2.7t^4$ | M1 |
Use $G''_Y(1) + G'_Y(1) - (G'_Y(1))^2$ | M1 |
$\text{Var}(Y) = 14.42 + 4.2 - 4.2^2 = 0.98$ | A1 |
---4 The discrete random variable $X$ has probability generating function $\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )$ given by
$$G _ { X } ( t ) = 0.2 t + 0.5 t ^ { 2 } + 0.3 t ^ { 3 }$$
The random variable $Y$ is the sum of two independent observations of $X$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability generating function of $Y$, giving your answer as an expanded polynomial in $t$. [3]
\item Use the probability generating function of $Y$ to find $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q4 [8]}}