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EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 1
Tutorial for Random Variables and Stochastic Processes
EEM2046 ENGINEERING MATHEMATICS IV
MULTIMEDIA UNIVERSITY
1. A coin is biased so that a tail is three times as likely to occur as a head. Find the
probability of the head appears when this coin is tossed once.
2. From a box containing 4 blue balls and 2 green balls, 3 balls are drawn in
succession, each ball being replaced in the box before the next draw is made.
(a) Find the probability distribution for the number of green balls.
(b) Express the probability distribution graphically as a probability histogram.
3. Suppose a discrete random variable X has the following probability mass function
( )3
10 ==XP , ( )
5
11 ==XP , ( )
4
12 ==XP , ( ) cXP == 3
where c is a constant. Find
(a) the value of c.
(b) ( )1≥XP .
4. Given a probability mass function of random variable X as follows:
( ) ( ) ( ) .3,2,1,0 ,8.02.033 == −
xCxfxx
xX
(a) Find ( )3=XP
(b) Find the cumulative distribution function, ( )xFX .
(c) Sketch the graph of cumulative distribution function, ( )xFX .
5. Let c be a constant and consider the density function ( )
<
≥=
−
0 if1
0 if1
2
2
yec
yec
yfy
y
Y.
(a) Find the value of c.
(b) Find the cumulative distribution function ( )yFY .
(c) Compute ( )1YF .
(d) Compute ( )5.0>YP .
6. Show that the function
( ) ( ).2,1;2,1,
18
2, ==
+= yx
yxyxf XY
satisfies the conditions of the joint probability distribution. Determine whether
the two random variables are independent or dependent.
7. Let the joint probability distribution for ( )321 XXX ,, be
( )( )
∞<<∞<<∞<<
=++−
elsewhere,0
0,0,0,,, 321
321
321
321
xxxexxxf
xxx
XXX .
(a) Find the marginal joint pdf for 1X and 2X .
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 2
(b) Find the marginal pdf for iX , 3,2,1=i , respectively.
(c) Find the conditional probability distribution for 21 , XX given 33 xX = .
8. Show that the two-dimensional random variables with the densities
( ) yxyxf XY +=, and ( ) ( )( )21
21, ++= yxyxg XY
if 10 ≤≤ x , 10 ≤≤ y , have the
same marginal distributions.
9. Let X and Y be the diameters (in cm) of a ball and a hole, respectively. Suppose
that X and Y has the joint probability density function
( ) ≤≤≤≤
=otherwise. 0
4.30.3 ,2.38.2 if25.6,
yxyxf XY
(a) Find the marginal distributions.
(b) What is the probability that a ball chosen at random will be able to put into a
hole whose diameter is 3.0?
10. Three balls are drawn without replacement from a box containing 3 green balls, 3
yellow balls, 6 black balls. Let X denote the number of yellow balls drawn and Y
the number of green balls drawn. Find
(a) the joint probability distribution function of X and Y.
(b) ( )2≥+YXP .
(c) the marginal distribution of X and Y, respectively.
(d) ( )12 =≥ YXP .
(e) Are X and Y independent?
11. Show that the random variables X and Y in joint probability distribution
( ) ∞<<<<
=−−
elsewhere0
002
,
,,,
yyxeyxf
yx
XY are statistically dependent.
12. Let joint probability distribution for random variables X and Y be
( ) ∞<<∞<<
=−−
elsewhere.,0
0,0,,
yxeyxf
yx
XY
Show that X and Y are statistically independent.
13. Let X be a random variable with probability distribution
( )
==
otherwise.,0
,5,4,3,2,1,5
1x
xf X
Find the probability distribution of random variable 15 −= XY .
14. Let X1 and X2 be discrete random variables with joint probability distribution
( )
==
=elsewhere.,0
,3,2,1;2,1,18, 21
21
2121
xxxx
xxf XX
Find the probability distribution of random variable 21XXY = . Lecture notes series, engineering mathematics vol. II
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 3
15. Let X be a continuous random variables with probability density function,
( )
≤<−=
elsewhere,0
42,6
xk
xf X .
Find k and the probability distribution of 2XY = .
16. Let X have the probability density function
( ) <<
=elsewhere0
101
,
, xxf X .
Find the probability distribution of lnXY 2−= .
17. Suppose X1 and X2 be two mutually independent random sample from Standard
Normal distribution, find the joint probability distribution of 2
2
2
11 XXY += and
22 XY = . Also find the marginal probability distribution of Y1.
18. Assume the waiting time of length X in minutes of a particular type of queue is a
random variable with probability density function
( )
>
=−
otherwise. ,0
0,6
16
1
xexf
x
X
(a) Determine the mean length of this type of queue.
(b) Find the variance and standard deviation of X.
(c) Find ( )26+XE .
19. Let ( ) ( ) ( ) ( ) ( )
=
=elsewhere0
1110003
1
,
,,,,,,,,
yxyxf XY . Find
−
−
3
2
3
1YXE .
20. If the joint probability density function of X and Y is given by
( ) ( )
<<<<+=
elsewhere0
21 1027
2
,
y,x,yxy,xf XY .
Find the expected value of ( ) YXY
XYXg 2
3, +
= .
21. The joint probability density function of random variables X and Y is
( )
<<<<+
=otherwise.,0
20 ,20,8,
yxyx
yxf XY
(a) Find the covariance of X and Y.
(b) Find the correlation coefficient of X and Y.
(c) Determine whether the X and Y are independent or dependent. Lecture notes series, engineering mathematics vol. II
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 4
22. Consider the joint density function
( )
<<>=
otherwise.,0
,10 ,2,16
, 3yx
x
y
yxf XY
Compute the correlation coefficient xyρ .
23. (a) Explain, very briefly, the method of least squares for obtaining the
equation of a regression line.
(b) The number of grams g of a certain detergent which will dissolve in 100g of
water at temperature 0C is shown in the table.
t(0C)
0
10
20
30
40
50
60
70
80
90
100
g(g) 53.5 59.5 65.2 70.6 75.5 80.2 85.5 90.0 95.0 99.2 104.0
Obtain the equation of the least squares regression line of g on t.
Estimate the value of g for the temperature of 450C.
24. A scientist working in agriculture research believes that there is a linear
relationship between the amount of certain food supplement given to hens and the
hardness of the shells of the eggs they lay. As an experiment, controlled
quantities of the supplement were added to the hens’ normal diet and the hardness
of the shells of the eggs then measured on a scale from 1 to 10, with the following
results:
Food supplement, f(g/day)
2 4 6 8
10
12
14
Hardness of the shells, g 3.2
5.2
5.5 6.4
7.2
8.5
9.8
(a) Find the equation of the regression line.
(b) Explain what the values of 1c and 2c tell you and why you should not try to
calculate the shell hardness for a food supplement of 20 g per day.
25. (a) A particle moves on a circle through points that have been marked 0, 1, 2, 3,
4 (in clockwise order). The particle starts at point 0. At each step, it has
probability 0.75 of moving the point clockwise (0 follows 4) and probability
0.25 of moving one point counterclockwise. Let Xn denote its location on the
circle after step n, find the one-step transition matrix.
(b) Suppose we have two boxes and 2d balls, of which d are black and d are red.
Initially, d of the balls are placed in box 1, and the remainder of the balls are
placed in box 2. At each trial a ball is chosen at random from each of the
boxes, and the two balls are put back in the opposite boxes. Let X0 denote
the number of black balls initially in box 1 and for n ≥ 1, let Xn denote the
number of black balls in box 1 after the nth trial. Find the transition function
of the Markov chain Xn, n ≥ 0.
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 5
26. Consider a Markov chain with transition matrix P with state space
{ }6,5,4,3,2,1=S as follows:
=
10
1
10
1
5
10
5
2
5
16
1
6
1
6
1
6
1
6
1
6
16
100
3
10
2
14
10
8
1
8
1
4
1
4
1
00004
3
4
1
00003
2
3
1
P .
Find
(a) ( )421P and ( )4
12P . Is ( )421P = ( )4
12P ? Give your reason.
(b) ( )421f . Explain the meaning of ( )4
21P and ( )421f .
27. Given a transition matrix with state space { }21, as follows:
3
1
3
22
1
2
1
(a) Sketch a state transition diagram.
(b) Calculate ( )312P , ( )3
21P , ( )21p and ( )22p given that ( )2
101 =p and ( )
2
102 =p .
Lecture notes series, engineering mathematics vol. II
28. Consider the following transition matrix with state space, { }210 ,,s = :
4
3
4
10
0012
1
2
10
.
Is the above transition matrix aperiodic and irreducible? If yes, after a long run,
what are the probabilities to be in states 0, 1 and 2?
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 6
29. Decompose the state space { }54321 ,,,,s = of the following transition matrix into
equivalent classes, then determine for every class whether they are recurrent or
transient.
(a)
100003
20
3
100
03
20
3
10
003
20
3
100001
(b)
100003
20
3
100
03
20
3
10
003
20
3
1
0003
2
3
1
30. Suppose a Markov Chain with state space, s ={1, 2} has transition
probability matrix as 1 , =+
= qp
pq
qpP .
(a) Find ( ) ( ) ( ) ( ) ( )511
4
11
3
11
2
11
1
11 , , , , fffff .
(a) Suppose initially the Markov Chain is in state 1, find ( )11 =XP and
( )21 =XP .
(b) Suppose initially the Markov Chain having the even chance to be in state 1
and state 2, find ( )11 =XP and ( )21 =XP .
31. Consider the Markov chain whose state transition diagram as below:
(a) Construct the transition matrix.
(b) Decompose the states into equivalent classes.
(c) Find the period for each of the state. Lecture notes series, engineering mathematics vol. II
0 1
3 2
21
31
21
54
41
32
51
43
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 7
PAST YEAR QUESTIONS
1. (a) Consider two random variables X and Y with joint probability density
function (pdf) as follows:
( ) ≤≤≤≤
=otherwise.0
10101
,
,y,x,y,xf XY
Determine the pdf of Z = XY.
(b) Consider a Markov chain with state { }3210 ,,, and transition probability matrix
=
10002
100
2
1
002
1
2
1
02
1
2
10
P
(i) Draw the state transition diagram.
(ii) Determine the recurrent and transient states.
(iii) Determine the period for recurrent states.
(iv) Find the probability that process goes from state 1 to 2 in 3 transitions. Final Examination, Trimester 3, Session 2001/02
2. (b) Suppose the transition probability matrix of a Markov chain with state
space {1, 2, 3, 4} is given as follows:
aaaa
acb
baba
ccba
4/0
2/2/2
(i) Find the values of a, b, and c.
(ii) Draw the state transition diagram.
(iii) Determine all the recurrent states.
(iv) Find the period for each of the recurrent state.
(v) Find the probability to go from state 1 to state 4 in 2 transitions.
(vi) Suppose we place a particle in state 1 at time 0, what is the probability
that the particle first reach state 4 at time 2. PEM2046 Final Examination(supp). Trimester 3, Session 2004/05.
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 8
Answer:
1. ( )41=HP .
2. (a)
X 0 1 2 3
P(X=x) 8/27 4/9 2/9 1/27
3. (a)6013=c (b)
32
4. (a) 0.008
(b) ( )
≥
<≤
<≤
<≤
<
=
.3 1,
,32 ,992.0
,21 0.896,
,10 0.512,
,0 ,0
x
x
x
x
x
xFX
5. (a) 4=c (b) ( )
≥−
<=
−0,1
0,
2
2
21
21
ye
yeyF
y
y
Y (c) 2
1
211
−− e (d) 4
1
21 −
e
6. X and Y are not independent.
7. (b) ( ) i
i
x
iX exf−= , 3,2,1 ,0 =∞<< ixi
9. (b) 0.5
10. (a)
( )yxf XY , x
0 1 2 3
y
0 111
449
1109
2201
1 449
11027
2209 0
2 1109
2209 0 0
3 2201 0 0 0
(b) 1/2
(c)
x 0 1 2 3
( )xf X 5521
5527
22027
2201
(d)
y 0 1 2 3
( )yfY 5521
5527
22027
2201
(e) 1/12. X and Y are not independent.
13. ( )
==
otherwise. 0,
24, 19, 14, 9, 4, ,5
1y
yfY
14.
( )wyg , y
1 2 3 4 6 x
1 181
182 0 0 0
2 0 182 0
184 0
3 0 0 183 0
186
( )yh 181
92
61
92
31
EEM2046 Engineering Mathematics IV Tutorial
Random Variables and Stochastic Processes Session 2010/11, trimester 3 9
15. ( )
≤≤
<<
=
otherwise. 0,
,164 ,12
1
,40 ,6
1
yy
yy
yfY
16. ( ) 0,2
2
>=−
ye
yf
y
Y
17. ( ) 12112
1
2
21
21 ,0,2
1,
1
yyyyeyy
yyfy
<<−∞<<−π
=−
;
( ) ∞<<=−
11 0,2
2
1
1y
eyf
y
Y
18. (a) 6 (b) 36 (c) 180
19. 1/9
20. 46/63
21. (a) 36
1−=σ XY (b)
11
1− (c) X and Y are not independent.
22. ( ) 4,E X =2 8
( ) , ( ) ,3 3
E Y E XY= = 0,xyσ = 0xyρ =
23. tg 4995.086.54 += . If 45=t , then Cg °= 34.77
24. (a) fg 5018.05286.2 +=
27. (a)
(b) ( )
72313
12 =P ,( )
54313
21 =P . ( )7241
1 2 =p , ( )7231
2 2 =p
28. 41
0 =π , 41
1 =π and 21
2 =π .
31. (a)
00
00
00
00
3
2
1
0
3210
51
54
43
41
32
31
21
21
(b) { }3 2, 1, 0,=C
(c) ( ) { } 2... 8, 6, 4, ,21 == d.c.gd . ( ) 2=id for 3 2, 1, 0,=i .
1 2