S3 June 2009 Q8
8. The random variable \(A\) is defined as
\[A = 4X - 3Y\]where \(X \sim \mathrm{N}(30, 3^2)\), \(Y \sim \mathrm{N}(20, 2^2)\) and \(X\) and \(Y\) are independent.
Find
The random variables \(Y_1\), \(Y_2\), \(Y_3\) and \(Y_4\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as
\[B = \sum_{i=1}^{4} Y_i\]| Scheme | Marks |
|---|---|
| \(\mathrm{E}(4X - 3Y) = 4\mathrm{E}(X) - 3\mathrm{E}(Y)\) | M1 |
| \(= 4 \times 30 - 3 \times 20\) \(= 60\) | A1 |
| (2) |
Notes
M1 for correct use of \(\mathrm{E}(aX + bY)\) formula
| Scheme | Marks |
|---|---|
| \(\mathrm{Var}(4X - 3Y) = 16\,\mathrm{Var}(X) + 9\,\mathrm{Var}(Y)\) 16 or 9; adding | M1; M1 |
| \(= 16 \times 9 + 9 \times 4\) \(= 180\) | A1 |
| (3) |
Notes
1st M1 for \(16\mathrm{Var}(X)\) or \(9\mathrm{Var}(Y)\)
2nd M1 for adding variances
Key points are the 16, 9 and +. Allow slip e.g using \(\mathrm{Var}(X) = 4\) etc to score Ms
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(B) = 80\) | B1 |
| \(\mathrm{Var}(B) = 16\) | B1 |
| \(\mathrm{E}(B - A) = 20\) \(\mathrm{E}(B) - \mathrm{E}(A)\) | M1 |
| \(\mathrm{Var}(B - A) = 196\) ft on 180 and 16 | A1ft |
| \(\mathrm{P}(B - A \gt 0) = \mathrm{P}\left(Z \gt \dfrac{-20}{\sqrt{196}}\right) = \left[\mathrm{P}(Z \gt -1.428\ldots)\right]\) stand. using their mean and var | dM1 |
| \(= 0.923\ldots\) awrt 0.923 – 0.924 | A1 |
| (6) | |
| (11 marks) |
Notes
1st M1 for attempting \(B - A\) and \(\mathrm{E}(B - A)\) or \(A - B\) and \(\mathrm{E}(A - B)\)
This mark may be implied by an attempt at a correct probability e.g. \(\mathrm{P}\left(Z \gt \dfrac{0 - (80 - 60)}{\sqrt{180 + 16}}\right)\). To be implied we must see the “0”
1st A1ft for \(\mathrm{Var}(B - A)\) can ft their \(\mathrm{Var}(A) = 180\) and their \(\mathrm{Var}(B) = 16\)
2nd dM1 Dependent upon the 1st M1 in part (c).
for attempting a correct probability i.e. \(\mathrm{P}(B - A \gt 0)\) or \(\mathrm{P}(A - B \lt 0)\) and standardising with their mean and variance.
They must standardise properly with the 0 to score this mark
2nd A1 for awrt 0.923 ~ 0.924