A2 June 2025 Q2
2. The discrete random variable \(X\) has probability distribution
| \(x\) | \(-1\) | \(a\) | \(b\) |
|---|---|---|---|
| \(\mathrm{P}(X = x)\) | \(\dfrac{1}{2}\) | \(\dfrac{1}{4}\) | \(\dfrac{1}{4}\) |
where \(a\) and \(b\) are positive constants.
The discrete random variable \(Y\) is defined as \(Y = a + bX\)
Given that \(\mathrm{Var}(Y) = \dfrac{1}{4}\mathrm{Var}(X)\) and \(\mathrm{E}(Y) = \dfrac{5}{16}\)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{E}(X) = -1 \times \dfrac{1}{2} + \dfrac{1}{4} \times a + \dfrac{1}{4} \times b\) [\(\mathrm{E}(X) =\)] \(\dfrac{1}{4}a + \dfrac{1}{4}b - \dfrac{1}{2}\) | M1 A1 | 1.1b 1.1b |
| (2) |
Notes
M1: Attempt at \(\mathrm{E}(X)\), at least two terms correct
A1: Full correct expression for \(\mathrm{E}(X)\), may be unsimplified
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{Var}(Y) = b^2\mathrm{Var}(X)\) | M1 | 1.1b |
| \(b^2 = \dfrac{1}{4} \quad \rightarrow \quad b = \dfrac{1}{2}\) | A1 | 1.1b |
| \(\mathrm{E}(Y) = a + b\mathrm{E}(X)\) or \(\begin{array}{|c|c|c|}\hline a - b & a + ab & a + b^2 \\ \hline 0.5 & 0.25 & 0.25 \\ \hline\end{array}\) | B1 | 2.1 |
| \(a + b\left(\dfrac{1}{4}a + \dfrac{1}{4}b - \dfrac{1}{2}\right) = \dfrac{5}{16}\) or \(\dfrac{1}{2}(a - b) + \dfrac{1}{4}(a + ab) + \dfrac{1}{4}(a + b^2) = \dfrac{5}{16}\) | M1 | 1.1b |
| \(a = \dfrac{4}{9}\) | A1 | 1.1b |
| \(\mathrm{E}(X) = \dfrac{1}{4}\left(\text{‘}\dfrac{4}{9}\text{’}\right) + \dfrac{1}{4}\left(\text{‘}\dfrac{1}{2}\text{’}\right) - \dfrac{1}{2}\) | M1 | 3.1a |
| \(\mathrm{E}(X) = -\dfrac{19}{72}\) | A1 | 1.1b |
| (7) | ||
| (9 marks) |
Notes
M1: Writing or using \(\mathrm{Var}(Y) = b^2\mathrm{Var}(X)\)
A1: For finding \(b = \dfrac{1}{2}\)
B1: Writing using \(\mathrm{E}(Y) = a + b\mathrm{E}(X)\), or writing a correct probability distribution for \(Y\)
M1: Correct expression for \(\mathrm{E}(Y)\) in terms of \(a\) and \(b\) (or their value of \(b\))
and equating their expression to \(\dfrac{5}{16}\)
A1: \(a = \dfrac{4}{9}\) or exact equivalent
M1: Substituting their values of \(a\) and \(b\) into their expression for \(\mathrm{E}(X)\) where \(a, b \neq \dfrac{5}{16}\)
A1: \(\mathrm{E}(X) = -\dfrac{19}{72}\) or exact equivalent