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1. A continuous random variable X has probability density function f(x) = It is known that P(X < 1) = 1 – . (a) Show that a = .

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IB Questionbank Mathematics Higher Level 3rd edition 1

1. A continuous random variable X has probability density function

f(x) =

 

, 0 .

e

0 ,

0 x a

x

ax

It is known that P(X < 1) = 1 – 2 1 .

(a) Show that a = ln 2 2

1 .

(6)

(b) Find the median of X.

(5)

(c) Calculate the probability that X < 3 given that X > 1.

(9) (Total 20 marks)

2. After a shop opens at 09:00 the number of customers arriving in any interval of duration t minutes follows a Poisson distribution with mean

10 t .

(a) (i) Find the probability that exactly five customers arrive before 10:00.

(ii) Given that exactly five customers arrive before 10:00, find the probability that exactly two customers arrive before 09:30.

(7)

(b) Let the second customer arrive at T minutes after 09:00.

(i) Show that, for t > 0,

P(T > t) = e

10

1 10

t

t

 

 

   .

(ii) Hence find in simplified form the probability density function of T.

(iii) Evaluate E(T).

(You may assume that, for n

+

and a > 0, lim e

 0

at n

t

t .)

(12) (Total 19 marks)

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