tut_RVSP_1011_3_

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 .



(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



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



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.



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?



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



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.



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



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

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