S2 January 2011 Q3
3. The continuous random variable \(X\) is uniformly distributed over the interval [−1,3].
Find
(a) \(\mathrm{E}(X)\) (1)
(b) \(\mathrm{Var}(X)\) (2)
(c) \(\mathrm{E}(X^2)\) (2)
(d) \(\mathrm{P}(X \lt 1.4)\) (1)
A total of 40 observations of \(X\) are made.
(e) Find the probability that at least 10 of these observations are negative. (5)
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = \dfrac{3 - 1}{2} = 1\) | B1 cao |
| (1) |
| Scheme | Marks |
|---|---|
| \(\mathrm{Var}(X) = \dfrac{(3 + 1)^2}{12} = \dfrac{4}{3}\) oe | M1A1 |
| (2) |
Notes
M1 \(\dfrac{(3 - 1)^2}{12}\) or \(\dfrac{(3 + 1)^2}{12}\) or \(\dfrac{(3 - -1)^2}{12}\)
A1 awrt 1.33
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X^2) = \dfrac{4}{3} + 1,\ = \dfrac{7}{3}\) oe | M1,A1 |
| (2) |
Notes
M1 “their(b)” + [“their (a)”]2 or \(\displaystyle\int_{-1}^{3} \frac{x^2}{4}\,\mathrm{d}x\)
A1 awrt 2.33
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \lt 1.4) = 0.6\) | B1 cao |
| (1) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}\left(X \lt 0\right) = 0.25\) \(Y\) is number of values less than 0 | B1 |
| \(Y \sim \mathrm{Bin}\left(40, 0.25\right)\) | M1A1 |
| \(\mathrm{P}(Y \geqslant 10) = 1 - \mathrm{P}(Y \leqslant 9)\) | M1 |
| \(= 1 - 0.4395 = 0.5605\) | A1 |
| (5) | |
| (11 marks) |
Notes
B1 For writing or using the probability of a negative = 0.25
M1 Writing or use of B(40, \(p\))
A1 Writing or use of B(40, 0.25)
M1 Writing or using \(1 - \mathrm{P}(Y \leqslant 9)\)
A1 awrt 0.561 or 0.560