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QQQ – S2 – Chapter 3 – Continuous Random Variables
[May 2013 Q5] 1.
The continuous random variable X has a cumulative distribution function

x  1,
 0,
 3
3x 2
 x
F(x) = 

 ax  b, 1  x  2,
 10 10
 1,
x  2,


where a and b are constants.
(a) Find the value of a and the value of b.
3 2
(b) Show that f(x) =
(x + 2x – 2), 1  x  2.
10
(c) Use integration to find E(X).
(d) Show that the lower quartile of X lies between 1.425 and 1.435.
[May 2012 Q5] 2.
density function
(1)
(4)
(3)
The queuing time, X minutes, of a customer at a till of a supermarket has probability
3
 32 x(k  x)

f(x) = 
0


(a)
(b)
(c)
(d)
(4)
0  x  k,
otherwise.
Show that the value of k is 4.
(4)
Write down the value of E(X).
(1)
Calculate Var (X).
(4)
Find the probability that a randomly chosen customer’s queuing time will differ from the mean by at
least half a minute.
(3)
[Jan 2012 Q6] 3.
A random variable X has probability density function given by
1
0  x  1,
 ,
2
1

f(x) =  x  ,
1  x  k,
2


0
otherwise,
where k is a positive constant.
(a) Sketch the graph of f(x).
1
(b) Show that k = (1 + √5).
2
(c) Define fully the cumulative distribution function F(x).
(d) Find P(0.5 < X < 1.5).
(e) Write down the median of X and the mode of X.
(f ) Describe the skewness of the distribution of X. Give a reason for your answer.
________________________________________________________________
Total: 42 marks.
Bronze: 29, Silver: 34, Gold: 38, Platinum: 42
(2)
(4)
(6)
(2)
(2)
(2)
QQQ – S2 – Chapter 3 – Continuous Random Variables
[May 2013 Q5] 1.
The continuous random variable X has a cumulative distribution function

x  1,
 0,
 3
3x 2
 x
F(x) = 

 ax  b, 1  x  2,
 10 10
 1,
x  2,


where a and b are constants.
(a) Find the value of a and the value of b.
3 2
(b) Show that f(x) =
(x + 2x – 2), 1  x  2.
10
(c) Use integration to find E(X).
(d) Show that the lower quartile of X lies between 1.425 and 1.435.
[May 2012 Q5] 2.
density function
(1)
(4)
(3)
The queuing time, X minutes, of a customer at a till of a supermarket has probability
3
 32 x(k  x)

f(x) = 
0


(a)
(b)
(c)
(d)
(4)
0  x  k,
otherwise.
Show that the value of k is 4.
(4)
Write down the value of E(X).
(1)
Calculate Var (X).
(4)
Find the probability that a randomly chosen customer’s queuing time will differ from the mean by at
least half a minute.
(3)
[Jan 2012 Q6] 3.
A random variable X has probability density function given by
1
0  x  1,
 ,
2
1

f(x) =  x  ,
1  x  k,
2


0
otherwise,
where k is a positive constant.
(a) Sketch the graph of f(x).
1
(b) Show that k = (1 + √5).
2
(c) Define fully the cumulative distribution function F(x).
(d) Find P(0.5 < X < 1.5).
(e) Write down the median of X and the mode of X.
(f ) Describe the skewness of the distribution of X. Give a reason for your answer.
________________________________________________________________
Total: 42 marks.
Bronze: 29, Silver: 34, Gold: 38, Platinum: 42
(2)
(4)
(6)
(2)
(2)
(2)
Answers