AS October 2020 Q3
3. The probability distribution of the discrete random variable \(X\) is
\[\mathrm{P}(X = x) = \begin{cases} \dfrac{k}{x} & \text{for } x = 1,\ 2 \text{ and } 3 \\[2mm] \dfrac{m}{2x} & \text{for } x = 6 \text{ and } 9 \\[2mm] 0 & \text{otherwise} \end{cases}\]
where \(k\) and \(m\) are positive constants.
Given that \(\mathrm{E}(X) = 3.8\), find \(\mathrm{Var}(X)\) (7)
| Scheme | Marks | AO |
|---|---|---|
| \(\Sigma p = 1 \rightarrow k + \tfrac{k}{2} + \tfrac{k}{3} + \tfrac{m}{12} + \tfrac{m}{18} = 1\) \(\Sigma px = 3.8 \rightarrow k + \tfrac{k}{2}(2) + \tfrac{k}{3}(3) + \tfrac{m}{12}(6) + \tfrac{m}{18}(9) = 3.8\) | M1 | 3.1a |
| \(\tfrac{11k}{6} + \tfrac{5m}{36} = 1\) [\(= 66k + 5m = 36\)] | A1 | 1.1b |
| \(3k + m = 3.8\) | A1 | 1.1b |
| Solving simultaneously to eliminate one variable | dM1 | 1.1b |
| \(k = \tfrac{1}{3}\) and \(m = \tfrac{14}{5}\) | A1 | 1.1b |
| \(\mathrm{E}(X^2) = 1^2 \times k + 2^2 \times \tfrac{k}{2} + 3^2 \times \tfrac{k}{3} + 6^2 \times \tfrac{m}{12} + 9^2 \times \tfrac{m}{18}\) [\(= 23\)] | M1 | 1.1b |
| \(\mathrm{Var}(X) = 23 - 3.8^2\) | ||
| \(= 8.56\) | A1 | 1.1b |
| (7 marks) |
Notes
M1: Attempt at both required equations with at least one term in \(k\) and one term in \(m\) correct
A1: Correct equation using \(\Sigma p = 1\)
A1: Correct equation using \(\Sigma px = 3.8\)
dM1: (dep on 1st M1) Solving simultaneously (may be implied by one correct value found)
A1: both values correct (may be implied by correct answer)
M1: Attempt to find \(\mathrm{E}(X^2)\) using their value of \(k\) and their value of \(m\) with at least 3 correct products or correct ft products Note: \(\mathrm{E}(X^2) = 6k + 7.5m\)
A1: 8.56 cao