S2 June 2013 (R) Q3
3. The random variable \(X\) has a continuous uniform distribution on \([a, b]\) where \(a\) and \(b\) are positive numbers.
Given that \(\mathrm{E}(X) = 23\) and \(\mathrm{Var}(X) = 75\)
Given that \(\mathrm{P}(X \gt c) = 0.32\)
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2}(a + b) = 23\) and \(\dfrac{1}{12}(b - a)^2 = 75\) | B1B1 |
| \(a + b = 46\) and \(b - a = \sqrt{12 \times 75}\ (= 30)\) | M1 |
| Adding gives \(2b = 76\) | M1 |
| \(\underline{b = 38}\) and \(\underline{a = 8}\) | A1 A1 |
| (6) |
Notes
alternative
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2}(a + b) = 23\) and \(\dfrac{1}{12}(b - a)^2 = 75\) | B1B1 |
| \(a + b = 46\) and hence \((46 - 2a)^2 = 900\) oe | M1 |
| \(a^2 - 46a + 304 = 0\) | |
| \((a - 8)(a - 38) = 0\) | M1 |
| \(\underline{b = 38}\) and \(\underline{a = 8}\) | A1 A1 |
| (6) |
1st B1 for at least one correct equation using given formulae
2nd B1 for any 2 correct equations for \(a\) and \(b\) using both 23 and 75
1st M1 for rearranging to get two linear equations in \(a\) and \(b\) or rearranging and substituting linear equation into quadratic.
2nd M1 for solving i.e. eliminating one variable leading to a linear equation in one variable or solving their quadratic correctly by any method.
1st A1 for \(b = 38\)
2nd A1 for \(a = 8\)
SC If they get \(b = 8\) and \(a = 38\) or they give two sets of values and do not eliminate one then they can get B1B1M1M1A1A0
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(23 \lt X \lt c) = 0.5 - 0.32\) or \(c = 28.4\) and prob \(= \dfrac{5.4}{30}\) | M1 |
| \(= \underline{0.18}\) | A1 |
| (2) | |
| (8 marks) |
Notes
M1 for a correct method, e.g. a correct expression or seeing calculation for \(c\) and calculation for probability
A1 for 0.18 only