S1 June 2017 Q4
4. The discrete random variable \(X\) has probability distribution
| \(x\) | \(-1\) | 0 | 1 | 2 |
|---|---|---|---|---|
| \(\mathrm{P}(X = x)\) | \(a\) | \(b\) | \(b\) | \(c\) |
The cumulative distribution function of \(X\) is given by
| \(x\) | \(-1\) | 0 | 1 | 2 |
|---|---|---|---|---|
| \(\mathrm{F}(x)\) | \(\dfrac{1}{3}\) | \(d\) | \(\dfrac{5}{6}\) | \(e\) |
| Scheme | Marks |
|---|---|
| \(a = \frac{1}{3}\) and \(e = 1\) | B1 |
| \(c = \left[1 - \tfrac{5}{6}\right] = \tfrac{1}{6}\) | B1 |
| \(\text{"}\dfrac{1}{3}\text{"} + 2b = \dfrac{5}{6}\) or \(\text{"}\dfrac{1}{3}\text{"} + 2b + \text{"}\dfrac{1}{6}\text{"} = 1\) | M1 |
| \(\Rightarrow b = \frac{1}{4}\) | A1 |
| \(d = a + b = \text{"}\tfrac{1}{3}\text{"} + \text{"}\tfrac{1}{4}\text{"}\) or \(d = \tfrac{5}{6} - \text{"}\tfrac{1}{4}\text{"}\) (o.e.) so \(d = \tfrac{7}{12}\) | B1ft |
| (5) |
Notes
Probabilities not in [0, 1] score 0 for corresponding A or B marks
Allow exact decimals or equivalent fractions
In part (a) you may see answers in the tables.
If answers in the table and answers on the page disagree take the answers on the page.
If jumbled working is followed by a list of answers on the page mark the list.
M1 for an equation for \(b\). Follow through their value of \(a\) and possibly \(c\) if both in [0,1]
Must be seen as an equation with \(b\) the only unknown.
NB \(b = d - a\) is not a suitable equation and use of this is M0
1st A1 for \(b = \frac{1}{4}\) or 0.25 (Correct answer only is 2/2)
3rd B1ft for \(d = \frac{7}{12}\) or their \(a\) + their \(b\) but their \(d\) must satisfy \(\frac{1}{3} \lt d \lt \frac{5}{6}\)
| Scheme | Marks |
|---|---|
| \(\left[\mathrm{P}(X^2 = 1) = a + b =\right]\ \dfrac{7}{12}\) | B1ft |
| (1) | |
| (6 marks) |
Notes
B1ft for \(\frac{7}{12}\) or their \(a\) + their \(b\) or their \(d\)
Please check the two B1ft marks carefully