The Hashemite University
Electrical Engineering Department
Fall 11-12
A
Probability: Final Exam َ◌(50%)
Instructor: Drs: A Al-Nimrat, A. Byati
Student Name:…………..
Serial number: …………..
Section:
Table (1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[30 points] Q1)Fill-in Table (1) above with the alphabet of the most correct answer for the following
questions:
1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. the probability
that the ticket drawn has a number which is a multiple of 3 or 5 is
a)
1
2
b)
8
15
c)
2
5
d)
9
20
e)None
2.A bag contains 2 red, 3 green, and 2 blue balls. Two balls are drawn at random. the
probability that none of the balls drawn is blue is
10
11
2
5
b)
c)
d)
e) None
a)
21
21
7
7
3. the probability of getting a sum 9 from two throws of a dice is
1
1
1
a)
b)
c)
6
8
9
d)
1
12
e) None
4. for a data entry job a candidate must take two exams; oral and written. If the probability to
pass the oral is 0.7, probability to pass the written is 0.5, and the probability to pass both is
0.25, then the probability that a candidate pass neither exam is
a) 0.45
b) 0.75
c) 0.95
d) 0.05
e) None
5. Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue
marbles. We randomly select 4 marbles from the bowl, with replacement. the probability of
selecting 2 green marbles and 2 blue marbles is
a) 0.135
b) 0.35
c) 0.432
1
d) 0.25
e) None
6. The average number of homes sold a company is 2 homes per day. the probability that
exactly 3 homes will be sold in 3 days is
a) 0.32
b) 0.026
c) 0.089
d) 0.45
e)None
7. Suppose scores on a test are normally distributed. If the test has a mean of 100 and a
standard deviation of 10, then the probability that a person who takes the test will score
between 90 and 110 (hint: F(1)= 0.84) is
a) 0.52
b) 0.68
c) 0.45
d) 0.78
e) None
8. If the probability that a student is absent in class is (0.02), then the probability that there
will be zero absence in a class of 40 students is
a) 1
b) 0.98
c)0.225
d)0.445
e)None
•
For the jpdf of X, Y
1
 xy
f X (x ) =  4
o
Answer (9,10,11)
9. P [X > 1,Y > 1] equals
5
9
a)
b)
16
16
c)
10. the correlation R X Y
5
16
a)
b)
16
9
2
2
11. the E [X +Y ] equals
c)
: 0 p X < 2, 0 p Y < 2
: o.w
3
8
1
8
d)
d)
16
5
1
8
e) None
e)None
a) 6
b) 2
c) 4
d) 8
e)None
• the jpdf of X, Y is
f X Y (x=
, y ) 0.12δ (x − 1)δ ( y − 1) + 0.18δ (x − 1)δ ( y − 2) + 0.28δ (x − 2)δ ( y − 1) + 0.42δ (x − 2)δ ( y − 2)
answer (12,13)
12. the correlation R X Y
a) 1.75
b) 2.72
c) 4.25
13. the mean value of X equals
a) 1.2
b) 0.7
c) 2.6
d) 3
e)None
d) 1.7
e)None
k (1 + x ) : 0 < x < 2
for f x (x ) = 
, answer (14,15)
: o.w
0
14. the value of K that makes f X (x ) a valid pdf equals
a) -0.25
b) 0.25
c) 0.5
d) 1.5
e)None
4
15. the E [X ] equals
•
a)
64
15
b)
32
21
c)
45
15
2
d)
5
7
e) None
[10 points] Q2) X ,Y are independent uniformly distributed random variables in [0,1] , if
=
Z (X +Y ) 2
Find:
a) The cumulative distribution function FZ (z ) .
b) The probability density function f Z (z ) .
[10 points] Q3) the jpdf of X ,Y is given by
1
: 0 p x < 2, 0 p y < 2, y < x
 (x + y )
f X (x ) =  4
o
: o.w
Find:
a) The marginal pdf f Y ( y ) .
b) The conditional density f X (x 0.5 < y < 1.5) .
3
The Hashemite University
Electrical Engineering Department
Spring 11-12
A
Probability: Final Exam (50%)
Instructor: Dr. A Al-Nimrat.
Student Name:…………..
Serial number: …………..
Table (1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 17 18 19 20 21 22 23 24 25
26
27
28
29
30
[30 points] Q1)Fill-in Table (1) above with the alphabet of the most correct answer for the following
questions:
1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. the probability
that the ticket drawn has a number which is prime number of is
a)
1
2
b)
8
15
c)
2
5
d)
9
20
e)None
2.A bag contains 2 red, 3 green, and 2 blue balls. Two balls are drawn at random without
replacement. the probability of the balls drawn are (G,B) is
11
5
6
6
a)
b)
c)
d)
e) None
21
42
21
7
3. The probability of getting even number from two throws of a dice is
1
1
1
3
a)
b)
c)
d)
e) None
12
8
9
12
4.In a box, there are 8 red, 7 blue, and 6 green balls. One ball is picked up randomly. the
probability that it is neither red nor green
1
3
8
7
b)
c)
d)
e) None
3
4
21
19
5.Three unbiased coins are tossed. the probability of getting at most two heads is
1
3
7
3
a)
b)
c)
d)
e) None
4
4
8
8
6.In a class, there are 15 boys and 10 girls. Three students are selected at random. The
probability that 1 girl and 2 boys are selected, is
a)
1
a)
21
46
b)
1
50
c)
25
117
d)
3
25
e) None
7. Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue
marbles. We randomly select 4 marbles from the bowl with replacement. the probability of
selecting 2 green marbles and 2 blue marbles is
a) 0.135
b) 0.35
c) 0.432
d) 0.25
e) None
8.From a pack of 52 cards, two cards are drawn together at random. the probability of both the
cards being queens is
1
25
35
1
a)
b)
c)
d)
e) None
15
57
256
221
9. The average number of homes sold by a company is 2 homes per day. the probability that
exactly 3 homes will be sold in 3 days is
a) 0.32
b) 0.026
c) 0.089
d) 0.45
e)None
10. Suppose scores on a test are normally distributed. If the test has a mean of 100 and a
standard deviation of 10, then the probability that a person who takes the test will score at
least a 100 is
b) 0.50
c) 0.45
d) 0.78
e) None
a) 0.52
11. If the probability that a student is absent in class is (0.02), then the probability that there
will be zero absence in a class of 40 students is
a) 1
b) 0.98
c)0.225
d)0.445
e)None
k (1 + x ) : 0 < x < 2
, answer (12,13)
for f x (x ) = 
: o.w
0
12. the value of K that makes f X (x ) a valid pdf equals
a) -0.25
b) 0.25
c) 0.5
d) 1.5
e)None
4
13. the E [X ] equals
•
64
32
45
5
b)
c)
d)
e) None
15
21
15
7
14.The probability that a student is accepted to a college is 0.3. If 5 students from the same
school apply, the probability that at most 2 are accepted is
a) 0.246
b) 0.580
c)0.837
d)0.345
e)None
a)
15. if X is N(0,1) and Y = X 2 then mean value of Y equals
a) 0
b) 1
c)0.5
d)-1
e)None
16. if the sample space of a random experiment is S = {0,1, 2.5,6} , if the random variable
X (s ) = cos(π s ) , the range of X is
a){-1,0,1}
b){-1,0}
c){-1,1}
d){0,1}
e) None
16. if X is uniform R.V. in [0,1] and Y = e − X , then f Y ( y ) equals
2
a) e − y u ( y )
d)
b)
1
[u ( y − e −1 ) − u ( y − 1)]
y
−1
[u ( y − e −1 ) − u ( y − 1)]
y
c) 1[u ( y ) − u ( y − 1)]
e)None
17. the mean value of Y in(16) equals
a) e −1 − 1
b)
e −1
e
c) e −1
d) 1 − e
e) None
18. if X is bernoulli R.V. with pmf : p [ X = 0] = q , p [ X = 1] = p , p + q = 1 , then the moment
generating function M X (s ) equals
a) (q + p ) n
b) (q + e s p )
d) ( p + e − s q ) n
c) ( p + e s q ) n
e)None
19. The characteristic function of R.V. X is Q X (ω=
) (q + e jw p ) n , then the variance of X
equals
a) − p
b) np
c) nq
d) npq
e) None
20. if X is uniform R.V. in [ 0, π ] , and Y = sin X ,then f Y ( y ) equals
a)
2
π 1− y
1
π 1+ y
2
2
[u ( y + 1) − u ( y − 1)]
[u ( y + 1) − u ( y )]
b)
d)
1
π 1+ y 2
2
π 1− y 2
[u ( y ) − u ( y − 1)]
[u ( y ) − u ( y − 1)]
c)
e) None
21. the pdf of R.V. X is f X=
( x ) 0.3δ ( x ) + 0.4δ ( x − 1) + 0.3δ ( x − 2) , if Y = X ! , then
FY ( y ) equals
a) 0.3u ( y ) + 0.7u ( y − 2)
b) 0.6u ( y − 1) + 0.4u ( y − 4)
c) 0.7u ( y − 1) + 0.3u ( y − 2)
d) 0.7u ( y ) + 0.3u ( y − 4)
22. The value of b that makes f X Y
a)
2
b) 4
e)None
x 2 + y 2 2
:x + y2 pb

, a valid jpdf is
(x , y ) =  π
0
:otherwise
c) 2
d) -4
e) None
π
 x   π
 y 
23. The value of b that makes FX Y ( x , y ) =
b  + tan −1     + tan −1    , a valid jcdf
 2   2
 2 
2
is
a)
π
b) 2π
c) π 2
d)
3
1
π2
e) None
24. The jpdf f X Y
 2 π

cos  xy  : −1 p x , y p 1
, then marginal pdf of X equals
(x , y ) = 
2

0
:otherwise

1

a)  sin π x  [u (x + 1) − u (x − 1) ]
π

1


b) 1 +
sin π x  [u ( x ) − u ( x − 1)]
 πx

c) [1 + sin π x ][u (x + 1) − u (x − 1) ]
1


sin π x  [u ( x + 1) − u ( x − 1)]
d) 1 +
 πx

e) None
25. the jcdf FX Y
(1 − e −α x )(1 − e − β y ) : x , y ≥ 0, α , β f 0
, then the marginal cdf of Y;
(x , y ) = 
:otherwise
0
FY ( y ) equals
b) (1 − e − β y )u ( y )
a) (1 − e − β y )u (− y )
d)
(−e − β y )u ( y )
d) (−e − β y )u ( y )
e)
c) u ( y )
e) None
26.the joint moment generating function of X,Y in (25) is
a) αβ e − (αs1 + β s2 )
b)
αβ
(α − s1 )( β − s 2 )
c) u ( y )
None
27. the correlation of X,Y R X Y in (25) equals
a) αβ
b) α
c)
1
(αβ ) 2
d)0
e) None
d)0
e) None
28. the covariance of X,Y C X Y equals
a) αβ
b) α
c)
1
(αβ ) 2
29.the correlation coefficient of X,Y ρ X Y in (25)equals
a) αβ
b) α
c)
1
(αβ ) 2
d)0
e) None
30. X,Y in (25) are
a)independent
b)uncorrelated
c) orthogonal
4
d)a+b
e)b+c
[10 points]
Q2) a random current is described by the sample space S = {−4 ≤ i ≤ 12} , if the
random varable X is defined as
 −2 : i ≤ −2
 i : −2 p i ≤ 1

X (i ) = 
1:1 p i ≤ 4
6 : i f 4
Find:
a) The range of R.V X.
b) What is the type of X.
c) The cdf FX ( x ) and plot it.
d) The pdf f X ( x ) and plot it.
5
[10 points]
0.5
Q3) the jpdf f X Y ( x , y ) = 
0
: 0 p x p 1,0 p y p 2
:otherwise
a) Find an expression for FX Y (x , y ) .
b) are X and Y independent? Justify your answer.
c) find the pdf f W (ω ) of W if W= X +Y .
6
The Hashemite University
Electrical Engineering Department
Spring 11-12
B
Probability: Final Exam (50%)
Student Name:…………..
Instructor: Dr. A Al-Nimrat.
Serial number: …………..
Table (1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 17 18 19 20 21 22 23 24 25
26
27
28
29
30
[30 points] Q1)Fill-in Table (1) above with the alphabet of the most correct answer for the following
questions:
1. In a class, there are 15 boys and 10 girls. Three students are selected at random. The
probability that 1 girl and 2 boys are selected, is
21
1
3
25
a)
b)
c)
d)
e) None
46
50
25
117
2. Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue
marbles. We randomly select 4 marbles from the bowl with replacement. the probability of
selecting 2 green marbles and 2 blue marbles is
a) 0.135
b) 0.35
c) 0.432
d) 0.25
e) None
3.From a pack of 52 cards, two cards are drawn together at random. the probability of both the
cards being queens is
1
25
35
1
a)
b)
c)
d)
e) None
15
57
256
221
4. The average number of homes sold by a company is 2 homes per day. the probability that
exactly 3 homes will be sold in 3 days is
a) 0.32
b) 0.026
c) 0.089
d) 0.45
e)None
5. Suppose scores on a test are normally distributed. If the test has a mean of 100 and a
standard deviation of 10, then the probability that a person who takes the test will score at
least a 100 is
a) 0.52
b) 0.50
c) 0.45
d) 0.78
e) None
6. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. the probability
that the ticket drawn has a number which is prime number of is
1
a)
1
2
b)
8
15
c)
2
5
d)
9
20
e)None
7.A bag contains 2 red, 3 green, and 2 blue balls. Two balls are drawn at random without
replacement. the probability of the balls drawn are (G,B) is
11
5
6
6
b)
c)
d)
e) None
a)
21
42
21
7
8. The probability of getting even number from two throws of a dice is
1
1
1
3
a)
b)
c)
d)
12
8
12
9
e) None
9.In a box, there are 8 red, 7 blue, and 6 green balls. One ball is picked up randomly. the
probability that it is neither red nor green
a)
1
3
b)
3
4
c)
8
21
d)
7
19
e) None
10.Three unbiased coins are tossed. the probability of getting at most two heads is
7
3
1
3
a)
b)
c)
d)
e) None
8
8
4
4
11. If the probability that a student is absent in class is (0.02), then the probability that there
will be zero absence in a class of 40 students is
a) 1
b) 0.98
c)0.225
d)0.445
e)None
k (1 + x ) : 0 < x < 2
for f x (x ) = 
, answer (12,13)
: o.w
0
11. the value of K that makes f X (x ) a valid pdf equals
a) -0.25
b) 0.25
c) 0.5
d) 1.5
4
12. the E [X ] equals
•
a)
64
15
b)
32
21
c)
45
15
e)None
d)
5
7
e) None
13.The probability that a student is accepted to a college is 0.3. If 5 students from the same
school apply, the probability that at most 2 are accepted is
a) 0.246
b) 0.580
c)0.837
d)0.345
e)None
14. if X is N(0,1) and Y = X 2 then mean value of Y equals
a) 0
b) 1
c)0.5
d)-1
e)None
15. the pdf of R.V. X is f X=
( x ) 0.3δ ( x ) + 0.4δ ( x − 1) + 0.3δ ( x − 2) , if Y = X ! , then FY ( y )
equals
a) 0.3u ( y ) + 0.7u ( y − 2)
b) 0.6u ( y − 1) + 0.4u ( y − 4)
2
c) 0.7u ( y − 1) + 0.3u ( y − 2)
d) 0.7u ( y ) + 0.3u ( y − 4)
16. The value of b that makes f X Y
a)
2
b) 4
e)None
x 2 + y 2 2
:x + y2 pb

, a valid jpdf is
(x , y ) =  π
0
:otherwise
c) 2
17. The value of b that makes FX Y
d) -4
e) None
π
 x   π
 y 
b  + tan −1     + tan −1    , a valid jcdf
(x , y ) =
 2   2
 2 
2
is
1
a)
π
18. The jpdf f X Y
c) π
b) 2π
 2 π
cos  xy
(x , y ) = 
2
0

2
d) π
2
e) None

 : −1 p x , y p 1 , then marginal pdf of X equals

:otherwise
1

a)  sin π x  [u (x + 1) − u (x − 1) ]
π

1


b) 1 +
sin π x  [u ( x ) − u ( x − 1)]
 πx

c) [1 + sin π x ][u (x + 1) − u (x − 1) ]
1


sin π x  [u ( x + 1) − u ( x − 1)]
d) 1 +
 πx

e) None
(1 − e −α x )(1 − e − β y ) : x , y ≥ 0, α , β f 0
19. the jcdf FX Y (x , y ) = 
, then the marginal cdf of Y;
0
:otherwise

FY ( y ) equals
b) (1 − e − β y )u ( y )
a) (1 − e − β y )u (− y )
c) u ( y )
d) (−e − β y )u ( y )
e) None
20. if the sample space of a random experiment is S = {0,1, 2.5,6} , if the random variable
X (s ) = cos(π s ) , the range of X is
a){-1,0,1}
b){-1,0}
c){-1,1}
d){0,1}
e) None
21.if X is uniform R.V. in [0,1] and Y = e − X , then f Y ( y ) equals
a) e − y u ( y )
d)
b)
1
[u ( y − e −1 ) − u ( y − 1)]
y
−1
[u ( y − e −1 ) − u ( y − 1)]
y
e)None
22. the mean value of Y in(16) equals
3
c) 1[u ( y ) − u ( y − 1)]
a) e −1 − 1
b)
e −1
e
c) e −1
d) 1 − e
e) None
23. if X is bernoulli R.V. with pmf : p [ X = 0] = q , p [ X = 1] = p , p + q = 1 , then the moment
generating function M X (s ) equals
a) (q + p ) n
d) ( p + e − s q ) n
c) ( p + e s q ) n
b) (q + e s p )
e)None
24. The characteristic function of R.V. X is Q X (ω=
) (q + e jw p ) n , then the variance of X
equals
a) − p
b) np
c) nq
d) npq
e) None
25. if X is uniform R.V. in [ 0, π ] , and Y = sin X ,then f Y ( y ) equals
a)
c)
2
π 1− y 2
1
π 1+ y 2
[u ( y + 1) − u ( y − 1)]
[u ( y + 1) − u ( y )]
b)
1
π 1+ y 2
2
d)
π 1− y 2
[u ( y ) − u ( y − 1)]
[u ( y ) − u ( y − 1)]
e) None
26.the joint moment generating function of X,Y in (19) is
a) αβ e − (αs1 + β s2 )
b)
αβ
(α − s1 )( β − s 2 )
c) αβ e (αs1 + β s2 )
d)
α − (α s + β s
e
β
1
2)
e) None
27. the correlation of X,Y R X Y in (19) equals
a) αβ
b) α
c)
1
(αβ ) 2
d)0
e) None
d)0
e) None
28. the covariance of X,Y C X Y in (19) equals
a) αβ
b) α
c)
1
(αβ ) 2
29.the correlation coefficient of X,Y ρ X Y in (19)equals
a) αβ
b) α
1
(αβ ) 2
c)
d)0
e) None
30. X,Y in (19) are
a)independent
b)uncorrelated
c) orthogonal
4
d)a+b
e)b+c
[10 points]
Q2) a random current is described by the sample space S = {−4 ≤ i ≤ 12} , if the
random varable X is defined as
 −2 : i ≤ −2
 i : −2 p i ≤ 1

X (i ) = 
1:1 p i ≤ 4
6 : i f 4
Find:
a) The range of R.V X.
b) What is the type of X.
c) The cdf FX ( x ) and plot it.
d) The pdf f X ( x ) and plot it.
5
[10 points]
0.5
Q3) the jpdf f X Y ( x , y ) = 
0
: 0 p x p 1,0 p y p 2
:otherwise
a) Find an expression for FX Y (x , y ) .
b) are X and Y independent? Justify your answer.
c) find the pdf f W (ω ) of W if W= X +Y .
6
The Hashemite University
Electrical Engineering Department
Spring 11-12
A
Probability: First Exam َ◌(25%)
Instructor: Dr. A Al-Nimrat.
Student Name:…………..
Serial number: …………..
Table (1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[30 points] Q1)Fill-in Table (1) above with the alphabet of the most correct answer for the following
questions:
1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. the probability
that the ticket drawn has a number which is a multiple of 3 or 5 is
a)
1
2
b)
8
15
c)
2
5
d)
9
20
e)None
2.A bag contains 2 red, 3 green, and 2 blue balls. Two balls are drawn at random. the
probability that none of the balls drawn is blue is
10
11
2
5
b)
c)
d)
e) None
a)
21
21
7
7
3. The probability of getting a sum 9 from two throws of a dice is
1
1
1
1
a)
b)
c)
d)
e) None
6
8
9
12
4.In a box, there are 8 red, 7 blue, and 6 green balls. One ball is picked up randomly. the
probability that it is neither red nor green
1
3
8
7
b)
c)
d)
e) None
3
4
21
19
5.Three unbiased coins are tossed. the probability of getting at most two heads is
1
3
7
3
b)
c)
d)
e) None
a)
4
4
8
8
6.In a class, there are 15 boys and 10 girls. Three students are selected at random. The
probability that 1 girl and 2 boys are selected, is
21
1
25
3
a)
b)
c)
d)
e) None
46
50
117
25
a)
1
7. Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue
marbles. We randomly select 4 marbles from the bowl with replacement. the probability of
selecting 2 green marbles and 2 blue marbles is
a) 0.135
b) 0.35
c) 0.432
d) 0.25
e) None
8.From a pack of 52 cards, two cards are drawn together at random. the probability of both the
cards being kings is
1
25
35
1
a)
b)
c)
d)
e) None
15
57
256
221
9. The average number of homes sold by a company is 2 homes per day. the probability that
exactly 3 homes will be sold in 3 days is
a) 0.32
b) 0.026
c) 0.089
d) 0.45
e)None
10. Suppose scores on a test are normally distributed. If the test has a mean of 100 and a
standard deviation of 10, then the probability that a person who takes the test will score
between 90 and 110 (hint: F(1)= 0.84) is
b) 0.68
c) 0.45
d) 0.78
e) None
a) 0.52
11. If the probability that a student is absent in class is (0.02), then the probability that there
will be zero absence in a class of 40 students is
a) 1
b) 0.98
c)0.225
d)0.445
e)None
k (1 + x ) : 0 < x < 2
for f x (x ) = 
, answer (12,13)
: o.w
0
12. the value of K that makes f X (x ) a valid pdf equals
a) -0.25
b) 0.25
c) 0.5
d) 1.5
e)None
4
13. the E [X ] equals
•
64
32
45
5
b)
c)
d)
e) None
15
21
15
7
14.The probability that a student is accepted to a college is 0.3. If 5 students from the same
school apply, the probability that at most 2 are accepted is
a) 0.246
b) 0.580
c)0.837
d)0.345
e)None
a)
15. if X is N(0,1) and Y = X 2 then mean value of Y equals
a) 0
b) 1
c)0.5
d)-1
e)None
16. if the sample space of a random experiment is S = {0,1, 2.5,6} , if the random variable
X (s ) = cos(π s ) , the range of X is
a){-1,0,1}
b){-1,0}
c){-1,1}
d){0,1}
e) None
[10 points] Q2) a random current is described by the sample space S = {−4 ≤ i ≤ 12} , if the
random varable X is defined as
2
 −2 : i ≤ −2
 i : −2 p i ≤ 1

X (i ) = 
1:1 p i ≤ 4
6 : i f 4
Find:
a) The range of R.V X.
b) What is the type of X.
c) The cdf FX ( x )
d) The pdf f X ( x )
Q3) the pdf of X is given by
1 − x2
f X ( x ) = e u ( x ) , if event A =
{1 p X ≤ 3}, B =
{X ≤ 2.5},C =A I B .
2
Find:
a) The cpdf f X ( x / B ) .
b) The probability of C.
3