A2 October 2020 Q4
4. The discrete random variable \(X\) has the following probability distribution.
| \(x\) | \(-5\) | \(-2\) | 3 | 4 |
|---|---|---|---|---|
| \(\mathrm{P}(X = x)\) | \(\dfrac{1}{12}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{4}\) | \(\dfrac{1}{2}\) |
The discrete random variable \(Y\) is defined in terms of the discrete random variable \(X\)
When \(X\) is negative, \(Y = X^2\)
When \(X\) is positive, \(Y = 3X - 2\)
| Scheme | Marks | AO |
|---|---|---|
| [\(\mathrm{E}(X) =\)] \((-5) \times \tfrac{1}{12} + (-2) \times \tfrac{1}{6} + (3) \times \tfrac{1}{4} + (4) \times \tfrac{1}{2}\ [= 2]\) | M1 | 1.1b |
| [\(\mathrm{E}(X^2) =\)] \((-5)^2 \times \tfrac{1}{12} + (-2)^2 \times \tfrac{1}{6} + (3)^2 \times \tfrac{1}{4} + (4)^2 \times \tfrac{1}{2}\ [= 13]\) (oe) | M1 | 1.1b |
| \(\mathrm{Var}(X) = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2 = 13 - 2^2 = \underline{\mathbf{9}}\) | A1 | 1.1b |
| (3) |
Notes
M1: Attempt at \(\mathrm{E}(X)\) with at least 3 correct products seen
M1: Attempt at \(\mathrm{E}(X^2)\) with at least 3 correct products seen
A1: 9 cao
Alternative
M1: Attempt at \(\mathrm{E}(X)\) with at least 3 correct products seen
M1: Attempt at expression for \(\mathrm{E}((X - \mu)^2) = (-5 - 2)^2 \times \tfrac{1}{12} + (-2 - 2)^2 \times \tfrac{1}{6} + (3 - 2)^2 \times \tfrac{1}{4} + (4 - 2)^2 \times \tfrac{1}{2}\) with at least 3 correct terms
A1: 9 cao
| Scheme | Marks | AO | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 | 3.1a | |||||||||||||||
| \(\mathrm{P}(Y \lt 9) = \mathrm{P}(X = -2) + \mathrm{P}(X = 3)\ \left[= \tfrac{1}{6} + \tfrac{1}{4}\right]\) | M1 | 1.1b | |||||||||||||||
| \(= \underline{\tfrac{5}{12}}\) | A1 | 1.1b | |||||||||||||||
| (3) |
Notes
M1: Finding distribution of \(Y\)
M1: \(\mathrm{P}(X = -2) + \mathrm{P}(X = 3)\) or \(\mathrm{P}(Y = 4) + \mathrm{P}(Y = 7)\)
A1: \(\tfrac{5}{12}\) (condone awrt 0.417)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{E}(XY) = (-5)(25)\tfrac{1}{12} + (-2)(4) \times \tfrac{1}{6} + (3)(7) \times \tfrac{1}{4} + (4)(10) \times \tfrac{1}{2}\) | M1 | 3.1a |
| \(= \underline{13.5}\) | A1 | 1.1b |
| (2) | ||
| (8 marks) |
Notes
M1: Attempt at \(\mathrm{E}(XY)\) with at least 2 correct terms
A1: 13.5