Download Faculty Of Computer Studies M130 Introduction to Probability and

Faculty Of Computer Studies
M130
Introduction to Probability and Statistics
FINAL EXAMINATION
Fall / 2013 – 2014
Number of Exam Pages
(including this cover sheet):
4 Time Allowed:
2 Hours
Keys
Instructions:
Please read the following instructions before starting:
1. This is a closed book exam.
2. This exam accounts to 50% of the total mark in the course.
3. Budget your time according to the mark assigned to each question.
4. Mobile phones and all other mobile communication equipments are
NOT allowed.
1
Part 1: MULTIPLE CHOICE QUESTIONS
You may solve all questions. Each question is worth 2 marks. Your
grade in this part is that of the best 5 questions, giving a total of 10
possible marks.
Q–1: A box contains 9 tickets numbered 1,2,…,9. If one ticket is drawn at
random, the probability that the number on the ticket is more than 3 is:
a) 1/2
b) 1/3
c) 2/3
d) 9/3
e) None of the above
Q–2: How many permutations are there of the letters of the word
STATISTICS?
10!
3!.3!
10!
b)
3!.3!.2!
a)
c) 10!
d)
10!
3!.2!
e) None of the above
Q–3: If a pair of fair dice is rolled, then the probability of getting a total of 6
is
a) 1/6
b) 7/36
c) 5/36
d) 12/36
e) None of the above
ax, 0  x  1
is a density function, then a =
0, otherwise
Q–4: If f x   
a)
b)
c)
d)
e)
2
0
1
-2
None of the above
Q–5: Which of the following can’t be the probability of an event?
2
a)
b)
c)
d)
e)
-4/7
0.0069
1
0
None of the above
Q–6: The following is known to be a probability distribution:
x
0
1
2
3
p(X=x)
?
0.25
0.35
0.20
The probability that x=0 is:
a) 0
b) 0.5
c) 1
d) 0.05
e) None of the above
4
0.15
Part 2: ESSAY QUESTIONS
Each question is worth 10 marks. You should answer only FOUR out of
the five questions, giving a total of 40 possible marks.
Q–1: [5+5 Marks] One bag contains 5 red balls and 4 black balls, and a
second bag contains 4 red balls and 6 black balls. One ball is drawn
from the first bag and placed unseen in the second bag.
a) What is the probability that a ball drawn from the second bag is
black?
p B2  
4 7 5 6 58
.  . 
9 11 9 11 99
b) What is the probability that the first ball is black given that a ball
drawn from the second bag is black?
4 7
.
P( B2  B1 ) 9 11 28
PB1 / B2  


58
P  B2 
58
99
Q–2: [2+2+2+4 Marks] Consider the experiment of tossing a coin 3 times.
Let the variable x be the number of tails.
3
a) Find the probability distribution for the number of tails.
x
0
1
2
3
p(X=x)=f(x) 1/8
3/8
3/8
1/8
b) Compute p(x  2)
2
P X  2    f  x  
1 3 3 7
  
8
8 8 8
x 0
c) Find the expected value for the number of tails.
3
1 3 3 1 3
E  X    xf x   0   1   2   3  
8 8 8 8 2
x 0
d) Find the variance and the standard deviation for the number of tails.
 2  E ( x 2 )  ( E ( x)) 2
 3   3   1  24
E ( x 2 )  0  1   4   9  
3
8 8 8 8
9 3
 2  3 
4 4
3

 0.866
4
Q–3: [3+2+2+3 Marks] Suppose x is a random variable with density
function given by:
1
 x 0 x2
f x    2
0
otherwise
a) Find the cumulative distribution function.
x
x
x
t 2 
1
x2
F ( x)   F (t )dt   t dt    
2
 4 0 4

0
if x  0
0
 2
x
Hence, F ( x)  
if 0  x  2
4


1 if x  2
b) Find p(x<1) and p(1<x<1.1)
4
p( x  1)  F (1) 
1
4
2

1.1
p(1  x  1.1)  F (1.1)  F (1) 
4

1
 0.0525
4
c) Find the mean of the random variable X.
2
 x3 
4
1 
  E  X    x x  dx    
2 
 6 0 3
0 
2
d) Find the variance of the random variable X.
 
E X2
2
 x4 
2 1 
  x  x  dx     2
2 
 8 0
0
2
 2  E X 2    2  2 
16 2

9 9
Q–4: [3+5+2 Marks]
a) Suppose x is a binomial random variable with n = 4 and p=1/3. Find
p(x=3)
3
1

 1   2  4! 1 2 8
p 3;4,  4C3     
. . 
3

 3   3  3!1! 27 3 81
b) A fair coin is tossed 8 times. Find the probability of getting at least 2
heads.
p( x  2)  1  p( x  2)
 1   p( x  0)  p( x  1)
 8  1  0  1  8  8  1  1  7 
 1            
 0  2   2   1  2  2  
9
247
 1

256 256
c) Find the coefficient of x 5 in the expression of 2 x  58 .
224,000
Q–5: [3+3+4 Marks] The weekly salaries of 5,000 employees of a large
corporation are assumed to be normally distributed with mean $450
and standard deviation $40.
Hint: use the below table.
5
Z
0.7
0.75
0.8
1
1.29
1.35
2
p(Z<z) 0.7580 0.7734 0.7881 0.8413 0.9015 0.9115 0.9772
a) If an employee is selected at random, find the probability that he or
she makes less than $480 .
P X  480   ?
x   480  450 3
z

  0.75

40
4
p( x  480)  p ( z  0.75)  0.7734
b) Find the probability that he or she makes between $480 and $530.
530  450
z1  0.75
z2 
2
40
p (480  x  530)  p (0.75  z  2)
 p ( z  2)  p ( z  0.75)
 0.9772  0.7734
 0.2038
c) Find the salary below which exist 90% of the employees’ salaries.
P( Z  z )  0.9 from the table z  1.29
Z
x

x  Z    40(1.29  450)  $501.6
6