| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.3 This is a straightforward application of normal distribution properties for linear combinations of independent random variables. Part (i) requires finding the distribution of X-Y and computing a probability, while part (ii) involves finding the distribution of 2X+2Y (perimeter). Both parts use standard techniques taught in A-level Further Maths statistics with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks |
|---|---|
| (ii) | X – Y ~ N(10, 5) |
| Answer | Marks |
|---|---|
| P (Perimeter > 75) = 0.943 | Either. cc or σ 2 : M1A0 |
| Answer | Marks |
|---|---|
| cwo | B1 |
Question 1:
--- 1 (i)
(ii) ---
1 (i)
(ii) | X – Y ~ N(10, 5)
8−10 11−10
z = [= –0.894…]; z = [= 0.447…]
1 2
5 5
P (8 < Breadth < 11) = 0.487
X 1 + X 2 + Y 1 + Y 2 ~ N(80, 10)
75−80
z= [= –1.58…]
10
P (Perimeter > 75) = 0.943 | Either. cc or σ 2 : M1A0
Both. FT on σ
cwo
As above
[e.g. N(80, 20), 0.8681: 2/4]
cwo | B1
M1A1
A1 [4]
B1
M1A1√
A1 [4]A machine is selecting independently and at random long rods and short rods. The length of the long rods, $X$ cm, is normally distributed with mean 25 cm and variance 3 cm$^2$ and the length of the short rods, $Y$ cm, is normally distributed with mean 15 cm and variance 2 cm$^2$. Assume that $X$ and $Y$ are independent random variables.
\begin{enumerate}[label=(\roman*)]
\item One long rod and one short rod are chosen at random. Find the probability that the difference in the lengths, $X - Y$, is between 8 cm and 11 cm. [4]
\item Two long rods and two short rods are chosen at random and are assembled into an approximately rectangular frame. Find the probability that the perimeter of the resulting frame is more than 75 cm. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2014 Q1 [8]}}