A2 June 2025 Q4
4. The discrete random variable \(X\) has probability generating function
\[\mathrm{G}_X(t) = \frac{1}{\left(1 - \dfrac{2t}{5}\right)} - k\]where \(k\) is a constant.
The random variable \(Y\) is defined as \(Y = 4X - 1\)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{G}_X(1) = 1 \quad \Rightarrow \quad \dfrac{1}{1 - \frac{2}{5}} - k = 1\) | M1 | 2.1 |
| \(k = \dfrac{2}{3}\) | A1 | 1.1b |
| (2) |
Notes
M1: For writing or using \(\mathrm{G}_X(1) = 1\)
A1: For finding \(k = \dfrac{2}{3}\) (must be exact)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{G}'_X(t) = -\dfrac{-\frac{2}{5}}{\left(1 - \frac{2t}{5}\right)^2}\) | M1 | 2.1 |
| [\(\mathrm{G}'_X(t) =\)] \(\dfrac{2}{5}\left(1 - \dfrac{2t}{5}\right)^{-2}\) may see [\(\mathrm{G}'_X(t) =\)] \(\dfrac{10}{(5 - 2t)^2}\) | A1 | 1.1b |
| \(\mathrm{G}'_X(1) = \dfrac{2}{5\left(1 - \frac{2}{5}\right)^2} \quad \Rightarrow \quad\) [\(\mathrm{E}(X) =\)] \(\dfrac{10}{9}\) | A1 | 1.1b |
| (3) |
Notes
M1: Attempt to differentiate \(\mathrm{G}_X(t)\) to obtain \(A\left(1 - \dfrac{2t}{5}\right)^{-2}\) (o.e.) for some constant \(A\)
A1: Correct differential (may be unsimplified)
A1: For finding [\(\mathrm{E}(X) =\)] \(\dfrac{10}{9}\) (must be exact, and must come from a correct differential)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{G}''_X(t) = (-2) \times \dfrac{2 \times \left(-\frac{2}{5}\right)}{5\left(1 - \frac{2t}{5}\right)^3} = \dfrac{8}{25\left(1 - \frac{2t}{5}\right)^3}\) or Use of binomial expansion or standard power series (at least two terms or \(t^2\) term correct) | M1 | 3.1a |
| Use of Maclaurin series for \(t^2\) (use of \(\mathrm{P}(X = 2) = \dfrac{\mathrm{G}''_X(0)}{2!}\)) or \(\dfrac{(-1)(-2)\left(-\frac{2t}{5}\right)^2}{2!}\) | M1 | 1.1b |
| \(\dfrac{1}{2} \times \dfrac{8}{25} = \dfrac{4}{25}\) or \(\dfrac{4}{25}\) | A1 | 1.1b |
| (3) |
Notes
M1: For \(\mathrm{G}''_X(t)\) in the form \(A\left(1 - \dfrac{2t}{5}\right)^{-3}\) or a correct expression (not value) for \(\mathrm{G}''_X(0)\)
Series method: Two terms correct from: \(\dfrac{1}{3} + (-1)\left(-\dfrac{2}{5}t\right) + \dfrac{(-1)(-2)}{2}\left(-\dfrac{2}{5}t\right)^2\)
M1: For correct method to find \(\mathrm{G}''_X(t)\) and use of \(\mathrm{P}(X = 2) = \dfrac{\mathrm{G}''_X(0)}{2!}\) (ft their \(\mathrm{G}'_X(t)\) in (b) provided \(\mathrm{G}'_X(t)\) not constant)
Series method: Correct unsimplified expression for \(t^2\) seen (may be part of expansion)
A1: For \(\dfrac{4}{25}\) o.e.
| Scheme | Marks | AO |
|---|---|---|
| Use of \(\dfrac{1}{t}\mathrm{G}_X(t)\) or \(\mathrm{G}_X(t^4)\) shown | M1 | 3.1a |
| \(\mathrm{G}_Y(t) = \dfrac{1}{t}\left(\dfrac{1}{1 - \frac{2t^4}{5}} - \text{‘}\dfrac{2}{3}\text{’}\right)\) | A1ft | 1.1b |
| (2) | ||
| (10 marks) |
Notes
M1: Sight or use of \(\dfrac{1}{t}\mathrm{G}_X(t)\) or \(\mathrm{G}_X(t^4)\) stated or effected in expression
A1ft: Correct expression, ft their \(k\) (accept \(k\) itself). ISW following a correct expression.