S2 January 2011 Q7
7. The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function
\[\mathrm{f}(x) = \begin{cases} kx(81 - x^2) & 0 \leqslant x \leqslant 9 \\ 0 & \text{otherwise} \end{cases}\]Using integration, find
Three independent customers shop at the post office.
| Scheme | Marks |
|---|---|
| \(\displaystyle\int_0^9 k(81x - x^3)\,\mathrm{d}x = 1\) | M1 |
| \(k\left[\dfrac{81}{2}x^2 - \dfrac{1}{4}x^4\right]_0^9 = 1\) | M1 |
| \(k\left(\dfrac{6561}{2} - \dfrac{6561}{4}\right) = 1\) \(k = \dfrac{4}{6561}\) **ag** | A1 cso |
| (3) |
Notes
M1 putting integral = 1 ignore limits. =1 must appear at least once in the working.
M1 attempting to integrate at least one part must have correct power of \(x\) (ignore limits)
A1cso subst of at least 9. Allow 1/1640.25
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = \displaystyle\int_0^9 kx^2(81 - x^2)\,\mathrm{d}x\) \(= k\left[\dfrac{81}{3}x^3 - \dfrac{x^5}{5}\right]_0^9\) | M1A1 |
| \(= k(19683 - 11809.8)\) | dM1 |
| \(= 4.8\) | A1 cao |
| (4) |
Notes
M1 attempt to use \(x\mathrm{f}(x)\) and attempt to multiply out bracket and attempt at integration – must have \(x^3\) and \(x^5\) terms (ignore limits)
A1 correct integration (ignore limits)
dM1 substituting correct limits (need not explicitly see 0). Dependent on having been awarded the first M1.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \gt 5) = \displaystyle\int_5^9 k(81x - x^3)\) | M1 |
| \(= k\left[\dfrac{81}{2}x^2 - \dfrac{1}{4}x^4\right]_5^9\) | M1d |
| \(= k\left(\dfrac{6561}{4} - 856.25\right) = \text{awrt } 0.478 \text{ or } \dfrac{3136}{6561}\) | A1 |
| (3) |
Notes
M1 attempting to integrate at least one part must have correct power of \(x\) (ignore limits)
M1 dep on previous M being awarded, substituting correct limits [may use \(1 - \displaystyle\int_0^5 k(81x - x^3)\) with limits 0 and 5]
| Scheme | Marks |
|---|---|
| P(At least 2 queue for more than 5 mins) \(= 3(1 - 0.478)(0.478)^2 + 0.478^3\) | M1A1ft |
| \(= 0.467\) | A1 |
| (3) | |
| (13 marks) |
Notes
M1 \(3(1 - p)p^2 + p^3\) or \(1 - (1 - p)^3 - 3(1 - p)^2 p\) 3 not needed
A1 for \(\mathbf{3}(1 - p)p^2 + p^3\) \(1 - (1 - p)^3 - \mathbf{3}(1 - p)^2 p\)
where \(p\) is their solution to part (c)
A1 awrt 0.467