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: 4]

Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.

(a)

Write down a suitable model for the distribution of the time, T minutes, that he has to wait for a train to leave.

[ 1 ]

(b)

Find the mean and standard deviation of T

During a 10-week period, Navtej travels to work by train on 46 occasions.

[ 3 ]

mean =3.5
standard deviation $=\sqrt{\frac{(7-0)^{2}}{12}}$ $=\frac{7}{\sqrt{12}}=2.0207 \ldots$

awrt 2.02

[Maximum number: 3]

In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per $\mathrm{m}^{2}$

Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per m^2

use a normal approximation to find the probability that in a randomly selected 20 m^2 of the peat bog there are fewer than 70 Common Spotted-orchids.

T\sim\mathrm{N}(90,90)

\mathrm{P}(T<70)=\mathrm{P}\left(Z<\frac{69.5-90}{\sqrt{90}}\right)=\mathrm{P}(Z<-2.160\ldots)=0.0154

B1 M1 A1

[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 ]

$\frac{5}{8}$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)