| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Challenging +1.2 Part (i) is a standard application of combining independent normal distributions (sum of means, sum of variances). Part (ii) requires setting up an inequality involving a linear combination (8C > T, where T = P + J + C), which becomes 7C > P + J, then applying properties of normal distributions. While this requires algebraic manipulation and understanding of linear combinations, it's a fairly routine Further Maths statistics problem with clear steps and no particularly novel insight required. |
| Spec | 5.04b Linear combinations: of normal distributions |
Total: 10 marks
**Part (i):**
$N(570, \ldots$ **M1**
$\ldots 470)$ **A1**
$1 - \Phi(1.845) = 0.0325$ **A1**
**Part (ii):**
$P(C > (P + J + C)/8)$ **M1A1** or equivalent e.g. $\frac{7}{8}C - \frac{1}{8}P - \frac{1}{8}J \sim N\!\left(-\frac{5}{4}, \frac{475}{32}\right)$
$= P(7C - P - J > 0)$ **M1**
$7C - P - J \sim N(-10, 950)$ **M1A1** SC: Common error: forgetting that $T(= P + J + C)$ and $C$ are **not** independent.
$P(> 0) = 1 - \Phi((0 - (-10))/\sqrt{950})$ **M1**
$= 1 - \Phi(0.3244) = \text{awrt } 0.373$ **A1** If $Y = k(8C - T)$ then $Y \sim N(-10k, 1110k^2)$ for non-zero $k$. Allow M1 A1. Then $P(Y>0)$ (or $P(Y<0)$ if $k<0$) $= 1 - \Phi((0-(-10k))/\sqrt{1110}) = 1 - \Phi(0.3002)$ M1dep $= \text{awrt } 0.382$ A1. Max 4/7.
Total: **10 marks**7 The total mass of a can of pears is the sum of three independent random variables: the mass of pears, the mass of juice, and the mass of the container. The mass in grams of pears in a can has the distribution $\mathrm { N } ( 300,400 )$. The mass in grams of juice has the distribution $\mathrm { N } ( 200,60 )$. The mass in grams of the container has the distribution $\mathrm { N } ( 70,10 )$.\\
(i) Find the probability that the total mass of a randomly chosen can is less than 530 g .\\
(ii) Find the probability that the mass of the container of a randomly chosen can is more than one eighth of the total mass of the can.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q7 [9]}}