S2 January 2007 Q5
5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha \lt x \lt \beta\).
(a) Write down the probability density function of \(X\), for all \(x\). (2)
(b) Given that \(\mathrm{E}(X) = 2\) and \(\mathrm{P}(X \lt 3) = \dfrac{5}{8}\) find the value of \(\alpha\) and the value of \(\beta\). (4)
A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
(c) \(\mathrm{E}(X)\), (1)
(d) the standard deviation of \(X\), (2)
(e) the probability that the shorter piece of wire is at most 30 cm long. (3)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(x) = \begin{cases} \dfrac{1}{\beta - \alpha}, & \alpha \lt x \lt \beta, \\ 0, & \text{otherwise.} \end{cases}\) | B1,B1 |
| (2) |
Notes
B1,B1 function including inequality, 0 otherwise
| Scheme | Marks |
|---|---|
| \(\dfrac{\alpha + \beta}{2} = 2, \quad \dfrac{3 - \alpha}{\beta - \alpha} = \dfrac{5}{8}\) | B1,B1 |
| \(\alpha + \beta = 4\) \(3\alpha + 5\beta = 24\) \(3(4 - \beta) + 5\beta = 24\) \(2\beta = 12\) \(\beta = 6\) | M1 |
| \(\alpha = -2\) | A1 |
| (4) |
Notes
B1,B1 or equivalent
M1 attempt to solve 2 eqns
A1 both
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = \dfrac{150 + 0}{2} = 75\text{ cm}\) | B1 |
| (1) |
Notes
B1 75
| Scheme | Marks |
|---|---|
| Standard deviation \(= \sqrt{\dfrac{1}{12}(150 - 0)^2}\) | M1 |
| \(= 43.30127\ldots\text{cm}\) | A1 |
| (2) |
Notes
A1 \(25\sqrt{3}\) or awrt 43.3
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \lt 30) + \mathrm{P}(X \gt 120) = \dfrac{30}{150} + \dfrac{30}{150}\) | M1,M1 |
| \(= \dfrac{60}{150}\) or \(\dfrac{2}{5}\) or 0.4 or equivalent fraction | A1 |
| (3) | |
| (12 marks) |
Notes
M1,M1 1st or at least one fraction, + or double