S2 June 2007 Q1
1. A string \(AB\) of length 5 cm is cut, in a random place \(C\), into two pieces. The random variable \(X\) is the length of \(AC\).
(a) Write down the name of the probability distribution of \(X\) and sketch the graph of its probability density function. (3)
(b) Find the values of \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). (3)
(c) Find \(\mathrm{P}(X \gt 3)\). (1)
(d) Write down the probability that \(AC\) is 3 cm long. (1)
| Scheme | Marks |
|---|---|
| Continuous uniform distribution or rectangular distribution. | B1 |
 | B1 B1 |
| (3) |
Notes
2nd B1 1/5, (0), 5 0 may be implied by start at \(y\) axis
3rd B1 for the shape: zero, horizontal line, zero
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = 2.5\) | B1ft |
| \(\mathrm{Var}(X) = \dfrac{1}{12}(5 - 0)^2\) or attempt to use \(\displaystyle\int_0^5 \mathrm{f}(x)x^2\,\mathrm{d}x - \mu^2\) | M1 |
| \(= \dfrac{25}{12}\) or 2.08 o.e | A1 |
| (3) |
Notes
B1ft ft from their \(a\) and \(b\), must be a number
M1 use their \(\mathrm{f}(x)\)
A1 awrt 2.08
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \gt 3) = \dfrac{2}{5} = 0.4\) | B1ft |
| (1) |
Notes
B1ft 2 times their 1/5 from diagram
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 3) = 0\) | B1 |
| (1) | |
| (8 marks) |