S1 June 2010 Q3
3. The discrete random variable \(X\) has probability distribution given by
| \(x\) | \(-1\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| \(\mathrm{P}(X = x)\) | \(\tfrac{1}{5}\) | \(a\) | \(\tfrac{1}{10}\) | \(a\) | \(\tfrac{1}{5}\) |
where \(a\) is a constant.
The random variable \(Y = 6 - 2X\)
| Scheme | Marks |
|---|---|
| \(2a + \tfrac{2}{5} + \tfrac{1}{10} = 1\) (or equivalent) | M1 |
| \(a = \dfrac{1}{4}\) or 0.25 | A1 |
| (2) |
Notes
M1 for a clear attempt to use \(\sum \mathrm{P}(X = x) = 1\)
Correct answer only 2/2.
NB Division by 5 in parts (b), (c) and (d) seen scores 0. Do not apply ISW.
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = \underline{1}\) | B1 |
| (1) |
Notes
B1 for 1
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X^2) = 1 \times \tfrac{1}{5} + 1 \times \tfrac{1}{10} + 4 \times \tfrac{1}{4} + 9 \times \tfrac{1}{5}\) (= 3.1) | M1 |
| \(\mathrm{Var}(X) = 3.1 - 1^2\), = 2.1 or \(\dfrac{21}{10}\) oe | M1 A1 |
| (3) |
Notes
1st M1 for attempting \(\sum x^2\mathrm{P}(X = x)\) at least two terms correct. Can follow through.
2nd M1 for attempting \(\mathrm{E}(X^2) - [\mathrm{E}(X)]^2\) or allow subtracting 1 from their attempt at \(\mathrm{E}(X^2)\) provided no incorrect formula seen.
Correct answer only 3/3.
| Scheme | Marks |
|---|---|
| \(\mathrm{Var}(Y) = (-2)^2\,\mathrm{Var}(X)\), = 8.4 or \(\dfrac{42}{5}\) oe | M1 A1 |
| (2) |
Notes
M1 for \((-2)^2\,\mathrm{Var}(X)\) or \(4\mathrm{Var}(X)\)
Condone missing brackets provided final answer correct for their Var(\(X\)).
Correct answer only 2/2.
Allow M1 for distribution of \(Y = 6 - 2X\) and correct attempt at \(\mathrm{E}(Y^2) - [\mathrm{E}(Y)]^2\) (this note is printed under (e) in the mark scheme)
| Scheme | Marks |
|---|---|
| \(X \geqslant Y\) when \(X\) = 3 or 2, so probability = “\(\tfrac{1}{4}\)” + \(\tfrac{1}{5}\) | M1 A1ft |
| \(= \tfrac{9}{20}\) oe | A1 |
| (3) | |
| (11 marks) |
Notes
M1 for identifying \(X\) = 2, 3
1st A1ft for attempting to find their P(\(X\)=2) + P(\(X\) = 3)
2nd A1 for \(\tfrac{9}{20}\) or 0.45