S1 June 2011 Q3
3. The discrete random variable \(Y\) has probability distribution
| \(y\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(\mathrm{P}(Y = y)\) | \(a\) | \(b\) | 0.3 | \(c\) |
where \(a\), \(b\) and \(c\) are constants.
The cumulative distribution function \(\mathrm{F}(y)\) of \(Y\) is given in the following table
| \(y\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(\mathrm{F}(y)\) | 0.1 | 0.5 | \(d\) | 1.0 |
where \(d\) is a constant.
| Scheme | Marks |
|---|---|
| \(a\) = 0.1 | B1 |
| [F(3) = F(2) + P(\(Y\) =3) =( 0.5 + 0.3)] \(d\) = 0.8 | B1 |
| \(b\) = F(2) - \(a\) = 0.5 - 0.1 or \(a + b = 0.5\) | M1 |
| \(b\) = 0.4 | A1 |
| \(c\) = 1 - F(3) or 1 - (\(a + b\) + 0.3) or \(a + b + c = 0.7\) \(c\) = 0.2 | A1 |
| (5) |
Notes
Correct answers with no (or irrelevant) working score full marks
1st B1 for \(a\) = 0.1
2nd B1 for F(3) = 0.8 or \(d\) = 0.8
M1 for a method for \(b\) or \(c\). E.g. sight of \(a + b = 0.5\) or \(a + b + c = 0.7\)
If their values satisfy one of these equations then score M1 provided their values are genuine probabilities (i.e. \(0 \lt p \lt 1\))
This M1 may be implied by a correct answer for \(b\) or \(c\)
1st A1 for \(b\) or P(2) = 0.4
2nd A1 for \(c\) or P(4) = 0.2 (corrected from the printed mark scheme: P(3) = 0.2)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(3Y + 2 \geqslant 8) = \mathrm{P}(Y \geqslant 2)\) or 1 - P(\(Y \leqslant 1\)) = \(b\) + 0.3 + \(c\) or 1 - \(a\) = 0.9 | M1 A1ft |
| (2) | |
| (7 marks) |
Notes
M1 for rearranging to P(\(Y \geqslant 2\)) or 1 - P(\(Y \leqslant 1\)) or selecting cases \(Y\) = 2, 3 and 4
A1ft for 0.3 + their \(b\) + their \(c\) or 1 - their \(a\), provided final answer < 1 and their values are probabilities.