Edexcel A-Level Mathematics A2 S2.3 Continuous Distributions Questions

Practise continuous uniform models and normal approximations, including density functions, interval probabilities and continuity corrections.

Exam points

state a uniform pdf over its interval and use interval length to calculate probabilities
choose a normal approximation and apply a continuity correction before standardising

Covered syllabus points

S2.3.1 - continuous uniform S2.3.2 - Normal approximation to the binomial distribution

[Maximum number: 10]

The random variable W has a continuous uniform distribution over the interval [-6, a] where a is a constant.

Given that $\operatorname{Var}(W)=27$

(a)

Show that a=12.

[ 2 ]

(b)

Given that $\mathrm{P}(W>b)=\frac{3}{5}$

[ 4 ]

(i)

find the value of b

[ 2 ]

(ii)

Find $\mathrm{P}\left(-12<W<\frac{b}{2}\right)$ .

[ 2 ]

$\mathrm{P}\left(-12<W<\frac{b}{2}\right)=\mathrm{P}(-6<W<0.6)=\frac{0.6+6}{18}=\frac{11}{30}$

(c)

A piece of wood AB has length 160 cm. The wood is cut at random into 2 pieces.
Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.

Calculate the probability that the area of the rectangle is greater than $975 \mathrm{~cm}^{2}$

[ 4 ]

$\frac{5}{8}$

[Maximum number: 5]

The manager of a supermarket is investigating the number of complaints per day received from customers.

A random sample of 180 days is taken and the results are shown in the table below.

Table for Question 2 — Edexcel A-Level Mathematics A2

A week consists of 7 consecutive days.

Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
Show your working clearly.
(Solutions relying on calculator technology are not acceptable.)

for writing or using N(21,21). May be seen in a standardisation expression.
for standardisation $( \pm)$ using their mean and sd. Allow 17.5, 18, 18.5, 19, 19.5, 22.5, 23, 23.5, 24,
24.5
for using $19 \pm 0.5$ or $23 \pm 0.5$
for a fully correct standardisation expression Implied by awrt $\pm 0.546$

A1*

awrt 0.291 or 0.293 from correct working seen

[Maximum number: 9]

Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15
- It always takes Navtej 3 minutes to walk to the bus stop
- Buses run every 15 minutes and Navtej catches the first bus that arrives
- Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work

The total time, T minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval $[\alpha, \beta]$

(a)

Show that $\alpha=32$

[ 1 ]

$3+[0]+29=32^*$

B1*

(b)

Show that $\beta=47$

[ 1 ]

$3+15+29=47^*$

B1*

(c)

State fully the probability density function for this distribution.

[ 2 ]

$f(t)=\begin{cases} \frac{1}{15}, & 32\le t\le 47,\\ 0, & \text{otherwise.} \end{cases}$

(d)

Find the value of

[ 3 ]

(i)

E(T)

[ 1 ]

(ii)

$\operatorname{Var}(T)$

[ 2 ]

\operatorname{Var}(T)=\frac{(47-32)^2}{12}=18.75

(e)

Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.

[ 2 ]

$(40-32)\times\frac{1}{15}$ $=\frac{8}{15}$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question 2

Question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question (i)

Question (ii)

Question (e)