1
MA2261 PROBABILITY AND RANDOM PROCESSES UNIT I RANDOM RANDO M VARIABLE
Random variables - Probability mass function ± Probability density functions- Properties ± Moments - Moment generating functions and their properties - Binomial, Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions and their properties - Functions of a random variables. -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
What is the need to define a random variable? In nature, the outcomes of many random experiments may be non-numerical. It is inconvenient to deal with these descriptive outcomes mathematically. For ease of manipulation, we assign a real number to each of the outcomes using a fixed rule or mapping. Define a random variable. Give an example. example . This rule or mapping from the original sample space (numerical or nonnumerical) to a numerical (real) sample space, subjected to certain constraints is called a random variable. Let S be the sample space of the random experiment. Random variable is a function whose domain is the set of outcomes and whose range is R, the real line. The random variable assigns a real value such that 1. The set
is an event for every
, for which a probability is defined.
Classify
random variables. A random variable can be classified as real random variable and complex random variable. Real random variable is one which takes on real values . A complex random variable Z is defined in terms of real variables X and Y as Z = X+iY. Define a discrete random variable. A random variable X is discrete if it assumes only discrete values (finite number or countably infinite number of possible discrete values). The sample space (domain) of discrete random variables may be discrete, continuous or a mixture of discrete and continuous points, but the co-domain is only discrete. Examples
1. The mark obtained by a student in an examination. Its possible values are 0, 40, 85 or 100. 2. The number of students who are absent for a particular period.
What is continuous random variable? A random variable X is said to be a continuous random variable if it takes all possible values between certain limits or in an interval which may be finite or infinite. Examples 1. The density of milk taken for testing at a farm. 2. The operating time between two failures of a computer.
What is a mixed random variable? Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
2
A random variable which can take both discrete and continuous values is called a mixed random variable. What is probability distribution function? If X is a real random variable, then the function defined by where is called distribution function of the random variable X. we also call it as cumulative distribution function. Give the properties of probability distribution function. function.
1. 2. 3. 4. 5.
What is probability density function? The probability density function
fined such that (i) (ii) (iii)
of a continuous random variable X is deand satisfies the following conditions
is integral over the range for all ,
Give the properties of probability density function? 1.
2. 3. 4. 5.
for all
Random
Probability Mass Function (pmf): For a discrete random variable pmf is
(i) (ii)
Probability Density Function (pdf):
variables
For a continuous random variable pdf is (i) (ii)
where
where
Cumulative
Distribution Function (c.d.f):
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
3
(i)
For discrete X :
(ii)
For continuous X:
for
Mathematical Expectation:
(i)
For discrete X :
(ii)
For continuous X :
Mean =
Variance =
Moment Generating Function:
(i)
For discrete X:
(ii)
For continuous X:
Characteristic
Function:
(i)
For discrete X:
(ii)
For continuous X:
Problems: 1. A one-dimensional random variable X has the following probability distribution.
0
1
2
3
4
5
6
7
0
2. A shipment of 6 television sets contains two defective sets. A hotel makes a random purchase of 3 of the sets. If X is the number of defective sets purchased by the hotel find (i)
Probability distribution of X
(ii)
Also determine its distribution function and hence, find the probability of purchasing at the most 1 defective set and probability of purchasing at least 1 defective set.
3. A continuous random variable X that can assume any value between
a density function given by
4. If (i)
. Find
Show that f(x) is a pdf Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
and
.
has
4 (ii)
Find its distribution function F(x)
5. If a random variable X takes the values 1,2,3,4 such that
. Find the probability distribution and cumulative
distribution function of X.
6. Given the probability distribution of X compute
X
-1
0
1
2
3
P(X)
0.15
0.1
0.3
0.3
0.15
7. A continuous random variable X has pdf
. Find the rth moment of X
about the origin. Hence find the mean and variance of X.
8. Find the MGF and rth moment for the distribution whose pdf is
. Find also standard deviation.
9. The first four moments of a distribution about X=4 are 1,4,10 and 45 respectively. Show that the mean is 5, variance is 3.
10. Let the random variable X have the pdf
find the moment generat-
ing function, mean and variance of X.
11. A 11. A continuous random variable X has the pdf C and CDF of X.
12. For the triangular distribution,
. Find the value of
find the mean, variance and the
moment generating function (MGF).
13. If a random variable has the MGF
, obtain the standard deviation of X.
14. The probability function of an infinite discrete distribution is given by
. Verify that the total probability is 1 and find the mean and variance
of the distribution. Find also,
.
15. If X is a random variable for which E(X)=10 and Var(X)=25. For what positive value of a and b, does
1)
have expectation 0 and variance 1?
If a If a random variable X takes the value 1,2 1, 2,3,4 such that 2P(X=1) 2P(X=1) = 3P(X= 3P(X=2 2) = P(X= P(X=3 3) = 5P( 5P(X= X=4 4). Find the
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
5 distribution of of x x 2)
A
random variable X has the f ollowing ollowing distribution, distribution,
X
0
1
2
3
4
5
6
7
P(X) P(X)
0
k
2k
2k
3k
k 2
2 k 2
7 k 2 +k
Find (i) the value of of k k (ii) P (1.5 < X < 4.5 | x > 2) (iii) the smallest value of o f P f or or which P(X P(X
½ (iv) f ind ind cdf cdf 3)
A
random variable X has the f ollowing ollowing distribution, distribution,
X
0
1
2
3
4
5
6
7
8
P(X) P(X)
a
3a
5a
7a
9a
11a
13a
15a
17a
Find (I) (I) the value of of aa (ii) P(X< P(X<3 3) 4)
A
(iii) cdf cdf
random variable X has the f ollowing ollowing distribution, distribution,
X
-2
-1
0
1
2
3
P(X) P(X)
0.1
k
0.2 0.2
2k
0.3 0.3
3k
Find (I) (I) the value of of k k (ii) P(X< P(X<2 2) & P (-2
6)
A
A
continuous R.V x has a PDF PDF given by f(x) f(x) = 3x ) < x < 1 Find µk¶such that P (X> k ) = 0.05
R.V has the PDF PDF
R.V has the PDF PDF
k ® ± f ( x ) ! ¯1 x 2 ± ° 0
A
8)
® ± PDF f(x) f(x) = ¯ A R.V has the PDF ± °
10)
if g
x g
determine µk¶ and the distribution f unction unction
otherwise
0 e x e 2 ®kx ±2k 2 e x e 4 ± f(x) f(x) = ¯ ±6k kx 4 e x e 6 ± elsewhere °0
7)
9)
(iii) cdf cdf MU MU
ind µk¶ µk¶ and cdf cdf f( x) f ind f(x)
x
0 e x e 1
2 x
ind its cdf cdf 2 e x e 4 f ind
x u 2
0
Find the distribution distribution f unction unction of of aa R.V x is given by F (x) = 1 ± ( ± ( 1+x) e-x; x >0 Find the den sity f unction, unction, P( x >2 >2)
The
cdf cdf o of the f the R.V x is given by
x 0
0 ® ± 2 x ± ¯ 2 ±1 (3 / 25)(3 x) ± 1 °
0 e x 1 / 2 1 / 2 e x 3
x u 3
of µk¶ µk¶ if if the the PDF PDF of of x x is f(x) f(x) = kx (1-x); 0 < x< 1. 11) Find the value of of the the f ollowing ollowing 12) Find the moment of X
0
2
3
4
6
F
3
7
2
3
5
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
Find P( |X| <1) and P( 1/3
6
6
13)
Let
±2
of the the a R.V. X with PDF PDF f(x) f(x) = 14) Find the MGF of
Let
16)
A
¡
± °0
other ise
x
¯2
f ind ind MGF
¢
0 x ¥
± £
15)
k 1,2,3,...
k have the probability mass f unction unction P(k) P(k) ¯ T k 2
¦
±0 °
x
1 x
¤
¥
¤
1 1 1 ind Q 1 , Q 2 2 also f ind
other ise §
X be a R.V with value -1, 0, 1 such that P(X= P(X= -1)=2P( -1)=2P(X=0) X=0) = P(X=1). P(X=1). Find the mean o of f 2 2x ± 5
continuous R.V X has the PDF PDF f(x) f(x) = kx2 e-x x > 0 Find the r th moment of of X X about the origin. Hence f ind ind the mean
and variance of of X X 17)
A
continuous R.V X has the PDF PDF f(x) f(x) = k e-x x > 0 Find the r th moment of of X X about the origin. Hence f ind ind the S.D S.D
18) For a R.V x with
M x (t ) !
(e t 2) 4 81
, Find P (x < 2)
distribution f or or which mean is 4 and variance 3 19) Determine the binomial distribution 729 times, how many times do you expect at least 3 dice to show 5 or 6? 20) 6 dice thrown 72 rom a f light light to hit a target. 21) 6 bombs are dropped f rom
The
probability of of hitting hitting is 1/ 1/5 . Two bombs are required to destroy
the building. Find the probability that the building is destroyed. of every 10 do not return. If 10 If 10 vessels are out , f ind ind the probability that at least 8 will ar22) In a long run, 3 vessels out of rive saf saf ely ely If X, Y are independent Poisson variable, then conditional distribution of of X X + Y given X is Binomial distribution. 23) If X, of binomial binomial distribution. 24) Poisson distribution is an approximation of ind the variance. If X is a Poisson variable such that P(X P(X = 2) = 9 P(X P(X = 4) + 90 P(X= P(X= 6) f ind 25) If X 26)
The
MGF of of aa RV of of X X is given by
M x ( t ) ! e 3 ( e
t
1)
Find P (x = 1)
o f the the books of of aa certain bindings have def def ective ective bindings. Find the probability that 2 of 100 f 100 books 27) It is known that 5% of bound of of this this binding will have def def ective ective 28)
A
certain rare blood type can be f ound ound in only 0.05% of o f people. If the population of of a randomly selected group is
3000. What is the probability that at least two people in the group have this rare blood type? 29)
A
radioactive source emits on the average 2.5 particles per second. Find th e probability that 3 or more particles will be
emitted in the interval of of 4 4 seconds. 30) A
radioactive source emits on the average 10 particles per min. in according to the Poisson law. Each particle emitted
has a probability of of 2/ 2/5 5 being recorded. Find the probability that 4 particles recorded in a min period 31)
From an arbitrary deck of of 5 52 cards, we draw cards at random with replacement and successively until an ace is drawn.
What is the probability that at least10 draw are needed 32) A
f ather ather asks his sons to cut their background lawn. Since he does not specif speci f y of of three sons is to do so the job, each
boy tosses a coin to determine odd person, what must cut lawn. In the case that all these get heads or tail, they continue tossing until they reach a decision. (i) f ind ind the probability that they reach a decision in less than µn¶ tosses (ii) what is minimum number of of tosses tosses required to reach a decision with probability 0.95 Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
7 33) A
woman and her husband want to have 95% chance f or or atleast one boy and atleast one girl. What is the minimum
number of of children children they should plan to h ave? assume that equal probability f or or gender of of child child 34)
lif lif e time of of IC chips manuf manuf actured actured by a semiconductor manuf manuf acturer acturer are approximately normally distributed with
mean 5x106 hours and variance 5x105 hours A mainf mainf rame rame manuf manuf acturer acturer requires at least 95% 95% of of aa batch should have a lif li f e time greater than 4x106 will the deal be made? 35) The
time required to repair a machine is exponentially distributed with parameter P=1/3 =1/3.. What is the probability that
the repair time exceeds 3 hours. 36) The The
daily consumption of of milk milk in excesses of of 20000 gallon i s approximately exponentially distributed with
U = 3000.
city has a daily stock of o f 3 35000 gallons. What is the probability that of of two two days selected at random the stock is in suf su f -
f icient icient f or or both days. 37)
the mileage which a car owner get with a certain kind of o f radial radial tyre is a R.V having exponential distribution with mean
40000 km. Find the probability that one of o f these these tyres will last 38)
If the time
T
1) at least 20000km
2) at most 30000km
is required to repair of of a component is exponentially distributed with P= ½ . What is the 1) probability
that repair time will exceed 2 hours
2) conditional probability that repair time takes atmost 10 hours given that its dura-
tion exceeds 9 hours? 39)
In a certain city, the daily c onsumptions of of electric electric power in millions of of kilowatt-hours kilowatt-hours can be treated as a R.V h aving
an Erlang distribution distribution with parameter P = ½; k = 3. If the If the power plant of of this this city has a daily capacity of o f 1 12 million kilowatt hours. What is the probability that this power supply will be inadequate on any given day?
FUNCTIONS OF RANDOM VARIABLE 40) The
41)
p.d.f p.d.f . of of x x be
®Pe P x x u 0 f ( x) ! ¯ °0 otherwise
forsome
P "0 3
Using method of of distribution distribution f unction unction calculate PDF PDF of of Y = x
42) Let
Y = e x f ind ind the p.d.f p.d.f . of of y y if x unif orm orm R.V. over ( over (0,1) f x is a unif
43) Let
Y = x 2 Find the p.d.f p.d. f . of of y y if if x x is a unif uni f orm orm R.V. over ( over (-1,2 -1,2)
44) The
45) Let
46)
p.d.f p.d.f . of of aa R.V x is f(x) f(x) = 2x ) < x < 1 Find the p.d.f p.d. f . of of 1) 1) Y = 3x + 1 2) Y = 8 X3
e x
¯ ¨
the p.d.f p.d. f . of of x x be f(x) f(x) = f ( x) !
x
other ise
0 °
±
Using the transf transf ormation ormation Y = X X
p.d.f o of Y f Y = ex W2 Find p.d.f
3
1 x
2 Find the p.d.f p.d. f . of of Y Y = ex & W = ( x-1)2 15 ± other ise 0 °
x be a R.V with p.d. p.d.f f f ( x ) !
4 x
0
©
If X If X is a normal R.V with mean zero and variance
47) Let
2
¯
48) Let
Mx(t) =
1 1 t
t < 1 be the m.g.f m.g. f o of the f the R.V. Find the m.g.f m.g. f . of of Y Y = 2X + 1
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
and Z = e ±x
8
49)
If x If x is unif uni f ormly ormly distributed in (
UNIT II J oint
T 2
,
T
2
p.d.f o of y = tan x ) Find the p.d.f
TWO DIMENSIONAL RANDOM VARIABLES
distributions - Marginal and conditional distributions ± Covariance - C orrelation and
regression - Transformation of random variables - Central limit theorem
--------------------------------------------------------------------------------------------------------------------------PART A 1.
Def ine ine joint probability density f unctions unctions of of aa 2-D random variables. variables.
2.
Def ine ine marginal density f unctions unctions of of aa 2-D random variables.
3.
Def ine ine conditional density f unctions unctions of of aa 2-D random variables.
4.
What is the condition f or or two variables to be independent. independent.
5.
Given joint pdf pdf , f(x , f(x ,y) = c x(x- y), y), 0 < x <2 , -x < y < x. x. Evaluate c.
6.
The
joint pdf pdf o of random variables x, y is given by f(x ,y) =
®8 x y, 0 x 1; 0 y x . ¯ ° 0, elsewhere
Find the marginal density
f unction unction of of x. x. 7.
If the If the joint pdf pdf o of a f a two dimensional random variables x,y is given by
®k (6 x y ); 0 x 2, 2 y 4 . elsewhere 0 , °
f(x ,y) = ¯
Find P( x < 1, y < 3).
8.
® ±3 ( x 2 y 2 ); 0 x 1, 0 y 1 f(x ,y) = ¯ 2 If f(x , f ind ind f(x f(x/y). ± 0, elsewhere °
9.
Find the value of of k k if if f(x,y) 1-x)(1-y) f or or 0 < x, y < 1 is to be a joint joint density densi ty f unction. unction. f(x,y) = k (1-x)(
PART B 10.
The
input to a binary communication system, denoted by random variable X, takes on one o f two values 0 or 1 with
probabilities ¾ and ¼ respectively respectively.. Because of of errors caused caused by noise in the system, the output Y diff diff ers ers f rom rom the input occasionally.
The
behaviour of the f the communication system is modeled by the conditional probability P( Y =1 / X =1) = ¾
and P ( Y =0 / x =0) = 7/ 7/8. Find (i) P (Y=1) (ii) P(Y=0) P(Y=0) and (iii) P( X =1/ =1/ Y =1).
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
9 11. 3 balls are drawn at r andom without replacement f rom rom a box containing 2 White , 3 Red & 4 Black balls. If X denots the number of of white balls drawn and Y denotes the number o f red balls drawn, f orm orm the joint probability probability distribution distribution of (X,Y).
12.
The
joint
probability
mass
f unction unction
of of X
Compute the marginal probability mass f unctions unctions X and Y.
and
Also,
Y
is
given
X Y
0
1
2
0
0 .1
0.04
.02
1
0.08
0. 2
0.06
2
0.06
0.14
0.3
as
.
f ind ind P( X e 1, y e 1) and check whether the variables are
independent. X / Y
13. Consider the discrete random variables X and Y with the joint pm f as f as shown below:
Are
X and Y independent?
2 1
1
0
1
1 / 16 1 / 16 1 / 16 1/ 8
1 / 16
1/ 8 .
1
1/ 8
1 / 16
1/ 8
2
1 / 16 1 / 16
1/ 8
un correlated? Are they uncorrelated?
14. If the joint pdf pdf o of a f a 2D rv (x,y) is given by f(x,y) f(x,y) =
x y ® ± x 2 ; ¯ 3 ± ° 0,
0 x 1, 0 y 2
.
elsewhere
Find (i) P ( X > ½ ) (ii) P ( Y < X) (iii) P ( Y< ½ / X < ½ ). 15.
The
joint pdf pdf o of random f random variables x, y is given by f(x by f(x ,y) =
®8 x y, 0 x 1; 0 y x . ¯ elsewhere 0 , °
(i) Find the conditional density f unctions unctions . (ii) Find P ( Y < 1/ 1/8 / X < ½ ) (iii) Check the independency. independency.
16.
6 ® ± ( x y 2 ); 0 e x e 1, 0 e y e 1 f(x ,y) = ¯ 5 If f(x , obtain the marginal densities of o f x x and y. ± 0, elsewhere °
Hence or otherwise f ind ind
17.
P ( ¼ e y e ¾ ).
2 ® ± ( x 2 y ); 0 x 1, 0 y 1 Given f(x ,y) = ¯ 3 , (i) ± 0, elsewhere °
f ind ind marginal densities of of X X and Y (ii) Conditional density X
given by Y =y and P ( X e ½ / Y = ½ ). 18. If f(x f(x ,y) =
®e ( x y ) , x u 0, y u 0 ¯ ° 0, elsewhere
, f ind ind (i) P (X < 1)
(ii) P ( X + Y < 1).
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
10 8
± x y , 1 x y 2
19.
The
joint density f unction unction of of 2 2 D random variables ( X, Y) is given by f(x ,y) =
¯9
. Find the the marmar-
± ° 0, else here
ginal density f unctions unctions of of X X and Y. Find also the conditional conditional density f unction unction of of Y Y given X=x and the conditional density f unction unction
of of
X given Y =y.
20.
® 9(1 x y) ; 0 e x g, 0 e y g ± pdf is is given by f(x ,y) = ¯ 2(1 x ) 4 (1 y ) 4 . The joint pdf ± elsewhere 0, °
Find the marginal marginal distributions distributions of of x x
and y and the conditional distribution distribution of of y y f or or X=x. 21.
The
Are
X and Y independent?
22. 22.
Let
joint density f unction unction is given by f(x,y) f(x,y) = 2, 0 < x < y < 1, f ind ind the marginal and conditional density f unction. unction.
the joint density f unction unction of of random variables X and Y be given by f(x,y) f(x,y) =
® ±1 y e x ; x " 0, 0 y 2 . ¯2 ± ° 0, elsewhere
Find
the marginal density f unctions unctions of of X X and Y. 23. 23. If the If the joint pdf pdf o of a f a two dimensional random variables x,y is given by
®6e 2 x 3 x f(x ,y) = ¯ ° 0, 1
2
;0
x1 ,
x 2
"0
elsewhere
. Find the probability tha t the f irst irst random variable will take on a value between 1
and 2 and the second random variable will take on a value between 2 and 3.
Also
f ind ind the probability that the f irst irst random
variable will take on a value less than 2 and the second random variable will take on a value greater than 2. 24. 24. If the If the joint pdf pdf o of a f a two dimensional random variables x,y is given by
®k (6 x y ); 0 x 2, 2 y 4 . elsewhere 0 , °
f(x ,y) = ¯
25.
(i) Find the value of of k. k. (ii) P ( X + Y < 3) (iii) P ( X < 1/ 1/ Y < 3 ).
® x y e x ; 0 x g, 0 y g ± pd f o of X f X and Y=y be given by f(x f(x/ y) = ¯ 1 y . Let the conditional pdf ± 0, elsewhere ° Find P(X P(X < 1 / Y < 2). CORRELATION AND REGRESSION PART A
26. Write the regression r egression equations. 27. Write the expression f or or acute angle between two lines of of regression. regression. 28. Prove that
-1e V e 1.
29. Prove that if if the the variables are independent , then they ar e uncorrelated. 30. If the If the variables are uncorrelated, are they independent? Justif Justi f y your answer. Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
11
PART B
31. Calculate the correlation coeff coeff icient icient f or or the f ollowing ollowing height (in inches) of of f f ather ( ather (X) and their sons( sons(Y)
32. 32.
X : 65
66
67
67
68
69
70
72
Y : 67
68
65
68
72
72
69
71.
The
joint probability mass f unction unction of of X & Y
/ Y
0 1
1 1
1 3
8 2
8 2
8
8
is given
below,
f ind ind the correlation coeff coeff icient icient
33. 33. If the If the joint density f unction unction of of (( X,Y) is given by f(x,y)= f(x,y)=2 2-x-y, o< x,y <1. Find correlation coeff coe ff icient icient 34. 34.
Let
X be a RV with mean value is 3 &Variance is 2. Find the second moment of of X X about the origin.
Another
RV Yis
def def ind ind by Y=-6x+22 Y=-6x+22.. Find the mean value of o f Y Y and the correlation of of X X &Y.
! 3 x y ( x y ),0 e x, y e 1.
35. If the If the joint p.d.f p.d. f o of ( f (X,Y) is given by f ( x , y ) Verif Verif y that E ( E ( Y
)
) ! E (Y ) !
11 24
of X&Y X&Y is given by 36. If the If the joint density of
37.
Two
independent
f ( x ) ! ¯
4 ax,
°0
,
random
0 x 1
x y ® ± , f ( x, y ) ! ¯ 3 ± ° 0, variables
other ise !
f ( y )
¯
!
0 x 1, 0 y
elsewhere
X
4by,
and
0
y
Y
1
other ise
°0,
2 .Obtain the r egression egression lines.
are
def ined def ined
such
that
. Prove that U = X+Y and V = X-Y are uncor-
!
related. 38. (X,Y) is a 2-D random variable unif unif ormly ormly distributed over the triangular region R bounded by y=0, x=3 x= 3 and y = 4x/3, /3, f ind ind the V x y. 39.
Let
the joint pdf pdf of (X,Y) be given by f ( x, y ) !
6 x y , x " 0, y " 0, x y
¯ "
#
1
else here
° 0,
.
Are
X and Y indepen-
$
dent? Obtain the regression lines. 40.
Let
the random variables X & Y have joint pdf pd f f ( x , y ) !
x y, 0
¯ %
° 0,
&
x
&
1, 0 y 1 &
&
else here
. Find the correlation
'
coeff coeff icient. icient. 41.
A
statistical investigator obtains the f ollowing ollowing regression lines 2x+3 x+3y = 5; 4y+3 y+3x = 7. Find (i) V x y
(iii) W y if
W x = 2.5
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
(ii) x, x , y
12 42. 42. Find V if f ( x , y )
!
24 y (1 x), 0 e y
e x e 1.
Find V x y.
43. 43. If x,y&z If x,y&z are un correlated with zero means , standard deviations 5,12 5,12,9 respectively and if i f u u = 2x -3 -3y, v= y + z + 2, f ind ind
Vu . v
44. 44. Given two random variables X and Y that have joint pdf pd f
f ( x, y ) !
x e ( x y ) ,
¯ (
°
x " 0, y " 0 else here
0,
. Find the
)
regression equation Y on X. 45. If the If the two dimensional random variable (X,Y) is unif unif ormly ormly distributed over R , where
R
! _( x, y ) / x 2 y 2 e 1, y u 0
coeff icient icient . a Find correlation coeff
TRANSFORMATIONS OF RANDOM VARIABLES PART A y 46. If X If X and Y are independent random variables having probability density f unctions f(x unctions f(x ) = e-x , x > 0 and f( and f( y) = e- y , y > 0, 0,
f ind ind the probability density f unction unction of of U = X+Y . 47. If X If X and Y are independent random variables having variances 2 and 3 respectively, f ind ind the variance of of 3 3x+4 x+4y. 48. If X If X and Y are independent random variables having identical unif unif orm orm distributions over (-1,1), f ind ind the density f uncunction X+Y. PART B
49.
The
joint pdf pdf o of X, f X, Y is given by f(x by f(x ,y) = e-(x+ y) , x > 0, y > 0. Find the pdf pd f o of u f u = (x+y)/ x+y)/ 2.
50. If X If X and Y are independent random variables each f ollowing ollowing N( N(0,2 0,2). Find pdf pd f o of z f z = 2x + 3y. 51.
an d Let X and
Y be positive independent random variables with identical pdf pd f ee-x , x > 0 and e- y y , y > 0. Find the joint pdf pdf
of U = X+Y and V = X / Y.
x y, 0 x, y 1 1
52. If the If the joint pdf pdf o of two f two random variables X and Y given by f ( x , y ) !
¯ 0
°0,
1
else here
. Find the the pdf pdf o of ( f (i) U =
2
XY (ii) U = X+Y 53. If X If X and Y each f ollow ollow exponential distribution with parameter 1 and are independent f ind ind the pdf pdf o of U = X ± Y. 54. If X If X and Y are independent random variables having densities f X ( x )
! e xU ( x), f ( y ) ! e yU ( y ) . Y
Find pdf pdf
of Z = X / Y. 55. If X If X and Y are independent random variables with identical unif uni f orm orm distributions in ( 0, a), f ind ind the density f unction unction of of Z = X ± Y. CENTRAL LIMIT THEOREM PART A
56. State the two diff diff erent erent f orms orms of of C Central limit theorem .
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
13 57. If Xi , i = 1 to 20 are independent ,unif ,unif ormly ormly distributed &identical variables , how are the random variables 20
X and
§ X
i
are distributed? distributed?
i !1
PART B
58. Prove Central limit theorem. 59.
A
random sample of of size size 100 is taken f rom rom population whose mean is 60 and variance 400. Using CLT what probabil-
ity can we assert that the mean of o f the the sample will not diff diff er f er f rom rom Q = 60 by more than 4 ? 60. If V If Vi, I = 1,2 1,2,«,2 ,«,20 are independent noise voltages received received in an µ adder µ and V is the sum of of the the voltages received, f ind ind the probability that total incoming voltage V exceeds exceeds 105 using CLT.
Assume
that each of of the the random variables Vi is
unif unif ormly ormly distributed over ( over ( 0, 10 ). 61.
A
distribution with unknown mean Q, has the variance equal to 1.5 . Use CLT to f ind ind how large a sample should be
taken f rrom om the distribution in order that the probability will be atleast 0.95 that the sample mean will be within 0.5 o f the f the population mean. 62. If X If X1,. X2,«.,Xn are Poisson variables with parameter P = 2. Use CLT, to estimate P(1 P(120 < Sn < 160 ) where Sn = X1 + X2+««+Xn and n = 75. UNIT III D efinition efinition
CLASSIFICATION OF RANDOM PROCESSES
and examples - first order, second o rder, strictly stationary, wide ± sense sta-
tionary and Ergodic processes - Markov p rocess - Binomial, Poisson and Normal processes Sine wave process ± Random Telegraphic Process. ------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Part A
1.
State the four types of stochastic processes
2.
Give an example for continuous time random process
3.
Define stationary process
4.
Give an example of Markov process
5.
Define Markov process
6.
When are the processes jointly WSS
7.
State the postulates of Poisson process
8.
Example for ergodic processes
Part B 9.
When is a stationary time series X (t) said to be ergodic? S . T it is ergodic for
Q if E[X (t)] = Q and cova-
riance function r(h) of X(t) satisfies r(h) p0 as hpg. 10. If the taxi arrive at a taxi stand from 2 directions independently according to the Poisson law with mean
rate of arrival 2 per 20 min and 4 per 15min. Find the probability that a person will have to wait for more than 5 min to catch a taxi Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
14 11. State the properties of ergodic process. 12. Define a random process. Explain the classification of Random Processes Give example 13. Derive the Poisson Law 14. Explain any two application of binomial process 15. Write a detailed note on sine-wave process 16. Two random process X(t) = A Cos Wt + B sin Wt & Y(t)= B cos Wt ² A sin Wt where A & B are uncorrelated
random variables with zero mean and equal variance and W is a constant, Check Whether the processes are jointly WSS 17. State and prove the properties of Poisson process 18. The
process
X(t)
whose
(at ) n 1
probability
under
certain
conditions
is
given
by
n ! 1,2,3,
(1 at ) n1 P { X (t ) ! n} ! (at )
S.T it is not a stationary
n!0
(1 at )
19. Suppose that a customer arrive at a bank according to Poisson process wit a mean rate 3 per min . Find the
probability that during a time interval of 2 min: a) Exactly 4 customers arrive b) More than 4 customers arrive. 20. Let {X (t) = a cos ( Wt+Y)} be a random process where Y and W are independent random variable . Further the characteristic function N of Y satisfies N(1) = N(2) = 0. While the density function f(w) of w satisfies f(w) = f(w). S.T X (t) is a WSS 21. Given that WSS random process X (t) = 10 cos (100t+U) where U is uniformly distributed over (- T,T). Prove that the process X (t) is correlation-ergodic . 22. Find the mean and autocorrelation of Poisson process. 23. If {X(t) = A cos Pt + B sin Pt; t u 0 } is a random process where A & B are independent N(0, W2) random variables, examine the stationary of X(t) . 24. Let {X(t); t u 0 }be a random process where X(t) = total number of points in the interval (0,t) = K,
1 ® !¯ ° 1
if
k
iseven
if
k
isodd
. Find ACF of X(t). Also if P(A=1)=P(A=-1) = ½ and A is independent of
X(t), find the power spectrum of Y(t) = A X(t) . 25. Two random process X(t) & Y(t) are defined by X(t) = A cos wt + B sin wt and Y(t) = B cos wt ² A sin wt . Show that X(t) and Y(t) are jointly WSS if A & B are uncorrelated random variables with zero means and the same variance and W is constant
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
15 26. Consider a random process X (t) = B cos (50t+U) where B and U are independent random variables . B is a random variable with mean zero and variance 1 U is uniformly distributed in the interval (- T,T). Find mean and autocorrelation of the process. 27. Distinguish SSS & WSS and establish any two results for weekly stationary time service involving auto correlation function. 28. Verify the sine wave process X(t) = Y coswt where Y is uniformly distributed in (0, 1) is SSS process. 29. S.T if a random process X(t) is WSS, then it must be covariance stationary. 30. Consider a R .P Y(t) = X(t) cos (wt+ U) where X(t) is WSS, U is random variable independent of X(t)and it is uniformly distributed (-T,T)and w is a constant. P.T Y(t) is WSS. 31. P.T the random process X(t) = A cos (wt+ U) where A,W are constant and U is uniformly distributed in (0,2 T) is correlation ergodic. UNIT IV
CORRELATION AND SPECTRAL DENSITIES
Auto correlation - Cross correlation - Properties ± Power spectral density ± Cross spectral density - Properties ± Wiener-Khintchine relation ± Relationship between cross power spectrum and cross correlation function
--------------------------------------------------------------------------------------------------------------------------Part A
)=e-2P|X|
R(X
1.
If
is a ACF of a process . Obtain S(w)
2.
Define auto correlation function. State all of its properties when the process is WSS
3.
Explain the concept of cross correlation function.
4.
Define cross spectral density
5.
State the properties of linear filter
6.
Describe a linear system with a random input .
7.
Example for cross-spectral density.
8.
Prove that S (w) is even when the process is real .
9.
Define spectral density and cross-spectral density.
10. what is mean by spectral analysis 11. Let X (t) = A Cos Pt + B sin Pt where A & B are independent normally distributed random variables
N (0, W2). Obtain R(X) 12. Let Xn = E Xn-1+ Yn for n= «.-1, 0, 1, 2, where E is a constant and {Y n} is a sequence of uncorrelated random
variables with zero mean and constant variance . Find R(X) 13. Compute R(X) of a random telegraph process taking values 5 & -4 with probabilities 1/3 & 2 /3 where the
no. of flips is according to Poisson process with mean rate of 2 per unit time . 14. Find the mean and autocorrelation of Poisson process.
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
16 15. Distinguish SSS & WSS and establish any two results for weekly stationary time service involving auto cor-
relation function. 16. Write a note on correlation integrals . 17. Explain the ACF of X·(t) in term of ACF of X(t) 18. The ACF for a stationary process X(t) is given by R xx(X) =
9
+2e
-|X |.
Find the mean and variance of Y=
2
´ X (t )dt and variance X(t). 0
19. If X(t) is a WSS with ACF Rxx(X) and if Y(t) = X( t+a) ² X( t-a), S .T
Ryy(X)= 2Rxx(X)- Rxx(X+2a)-Rxx(X-2a). 20. Consider the two random process X(t) = 3 cos (wt + U) and Y(t) = 2 cos (wt + U -T/2) where U is a random variable uniformly distributed in (0,2 T) P.T
Rxx( Rxx(0)Ryy( Ryy(0) u Rxy( Rxy(X )
21. Given R(X) = ce-E|X| Obtain the spectral density Also state its application to linear system with random inputs 22. Let {X (t) = a cos ( Wt+Y)} be a random process where Y and W are independent random variable . Further the characteristic function N of Y satisfies N(1) = N(2) = 0. While the density function f(w) of w satisfies f(w) = f(w). S.T X (t) is a WSS and calculate the spectral density. 23. State and establish the spectral representation theorem for WSS process. Also, obtain the covariance function and spectral devising function average process . 24. Describe a linear system. With usual notation S.T S yy(w) = Sxx(w) | H (w) | 2 where Syy(w) & Sxx(w) are the spectral density function of Y(t) and X(t) and H(w) is the system transfer function . 25. Write a note on correlation integrals . 26. Write a note on linear system with random inputs . 27. Given the ACF of X(t) = A e -a|X|cos wX where A > 0 , a > 0 and w are real constant . Find S(w) 28. A system has an impulse response h(t) = e ²at U(t) find the power spectral density of the output Y(t) corresponding to the input X(t).
29. Given that S XY (Z )
®a ibZ , Z 1 !¯ else where °0 UNIT V
find its cross-correlation function
.
LINEAR SYSTEMS WITH RANDOM INPUTS
Linear time invariant system - System transfer function ±Linear systems with random inputs ± Auto correlation and cross c orrelation functions of input and output
---------------------------------------------------------------------------------------------------------------------------
Dr. D. Saravanan, Saravanan, Professor of Mathematics Mathematics
