CAIE A-Level Mathematics A2 6.2 Linear Combinations of Random Variables Questions

Practise forming distributions for sums, differences and scaled independent normal variables before calculating probabilities.

Exam points

combine means linearly but add squared-coefficient variances for independent variables
translate a contextual comparison into one linear combination before standardising
use the derived mean and standard deviation to calculate the required probability

Covered syllabus points

6.2.1—Normal distribution

[Maximum number: 5]

The masses in kilograms of large and small bags of cement have the independent distributions N(50,2.4) and N(26,1.8) respectively.

Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement.

$D\sim N\bigl(5\times50-10\times26,\;5\times2.4+10\times1.8\bigr)=N(-10,30)$ $z=\frac{0-(-10)}{\sqrt{30}}=1.826$ $P(D>0)=1-\Phi(1.826)=0.0339\approx0.0340$

Award marks for the distribution, variance, standardisation and tail probability;
final answer 0.0339 or 0.0340 to 3 significant figures.

[Maximum number: 3]

The independent random variables X and Y have the distributions $\operatorname{Po}(1.9)$ and $\operatorname{Po}(2.2)$ respectively．

Find P(X+Y<4) ．

$\lambda=1.9+2.2[=4.1]$ $e^{-4.1}\left(1+4.1+\frac{4.1^{2}}{2!}+\frac{4.1^{3}}{3!}\right)$

or $\mathrm{e}^{-4.1}(1+4.1+8.405+11.487)$ or 0.01657+0.06795+0.13929+0.19037
Marking guidance:

Allow any $\lambda$ .

Allow one end error.

Must see expression.

=0.414(3 sf)
SC: unsupported answer 0.414 scores B1 B1.

[Maximum number: 3]

A bus station has exactly four entrances. In the morning the numbers of passengers arriving at these entrances during a 10 -second period have the independent distributions $\operatorname{Po}(0.4), \operatorname{Po}(0.1), \operatorname{Po}(0.2)$ and Po(0.5).

Find the probability that the total number of passengers arriving at the four entrances to the bus station during a randomly chosen 1 -minute period in the morning is more than 3.

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Covered syllabus points

Question 1

Question 2

Question 3