| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 11 |
| Topic | The Gamma Distribution |
| Type | Computing expectation from pdf |
| Difficulty | Standard +0.3 This is a straightforward application of standard probability density function properties. Students must use the given integral formula to find constants from the mean condition, compute variance using E(T²) - [E(T)]², and evaluate a probability using integration by parts. While it involves multiple parts and some algebraic manipulation, all steps follow routine procedures with the integral result provided, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
**Question 11**
**(i)**
$$\int_0^\infty at\mathrm{e}^{-bt}\,\mathrm{d}t = 1 \Rightarrow a \times \frac{1}{b^2} = 1$$ B1
$$\int_0^\infty at^2\mathrm{e}^{-bt}\,\mathrm{d}t = \frac{3}{2} \Rightarrow a \times \frac{2}{b^3} = \frac{3}{2}$$ B1
Hence $\dfrac{2}{b} \times \dfrac{a}{b^2} = \dfrac{3}{2} \Rightarrow b = \dfrac{4}{3}$ (AG) B1
$a = b^2 = \dfrac{16}{9}$ B1 **[4 marks]**
**(ii)**
$$\mathrm{E}(T^2) = \int_0^\infty at^3\mathrm{e}^{-bt}\,\mathrm{d}t = a \times \frac{6}{b^4}$$ M1
$$= \frac{16}{9} \times 6 \times \left(\frac{9}{16}\right)^2 = \frac{27}{8}$$ A1
$$\mathrm{Var}(T) = \frac{27}{8} - \frac{9}{4} = \frac{9}{8}$$ A1 **[3 marks]**
**(iii)**
$$\int_0^{1.5} \frac{16}{9}t\mathrm{e}^{-\frac{4}{3}t}\,\mathrm{d}t = \left[-\frac{4}{3} \times \frac{16}{9}t\mathrm{e}^{-\frac{4}{3}t}\right]_0^{1.5} + \int_0^{1.5}\frac{16}{9} \times \frac{3}{4}\mathrm{e}^{-\frac{4}{3}t}\,\mathrm{d}t$$ M1
$$= -\frac{4}{3} \times \frac{3}{2}\mathrm{e}^{-2} - 0 + \left[-\frac{3}{4} \times \frac{4}{3}\mathrm{e}^{-\frac{4}{3}t}\right]_0^{1.5}$$ A1
$$= -2\mathrm{e}^{-2} - \mathrm{e}^{-2} + 1 = 1 - 3\mathrm{e}^{-2} \approx 0.594$$ A1
Hence median $<$ mean since $0.594 > 0.5$ A1 **[4 marks]**11 The time, $T$ years, before a particular type of washing machine breaks down may be taken to have probability density function f given by
$$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $b$ are positive constants. It may be assumed that, if $n$ is a positive integer,
$$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$
(i) Records show that the mean of $T$ is 1.5 . Show that $b = \frac { 4 } { 3 }$ and find the value of $a$.\\
(ii) Find $\operatorname { Var } ( T )$.\\
(iii) Calculate $\mathrm { P } ( T < 1.5 )$. State, giving a reason, whether this value indicates that the median of $T$ is smaller than the mean of $T$ or greater than the mean of $T$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q11 [11]}}