15MA209_May2017-4_Sem.pdf

(1)

Reg. No.

B.Tech.

DEGREE

EXAMINATION,

MAY

2017 Third / Fourth Semester

15MA2O9 _

PROBABILITY AND RANDOM

PROCESS

(For the candidates admitted during the academic year 2015

-

2016 onwards)

'

Note:

(0

Part -

A

should be answered in OMR sheet within

frst

45 minutes and OMR sheet should be handed

over to hall invigilator at the end of45m minute.

'

(iD

Part - B and Part - C should be answered in answer booklet.

Time: Three

Hours

Max.

Marks:

100

PART-A(20x1=20Marks)

Answer

ALL

Questions

1.

The distribution function

of

a random variable

X

is given

by

F(x)=1-(1a*)e-*,x>0

then

its density function is

(A) (t+r)r--

+e-',x)o

(B)

xe-',x>o

(C)

xe',x>O

@) I

,-,.,

> o

l+x

2.

The

random variable

X

has

a binomial distribution with

parameters

n:20,

p:0.4

then

its

mean is

(A)

I

(B)

80

(c)

0.8

(D)

50

3. If var(x)

= 4 then find

var(4X

+ 5)where

X

is a random variable

(A)

64

(B)

16

(c)

21

(D)

6e

4.

If

F is the distribution fimction of the random variable

X

and

if

a<b then

P(a

<

X

<b)

-(t) p(x-a)+r(o)-r(a)

(B)

F(6)-F(a)-r(x-b)

(C)

p(a<x<b)+

p(x=a)

@)

F(D)-F(a)

5

.

If

F(x,

y)

is the

cdf

of a two dimensional continuous random variable, then

f(x, y)

is obtained from

fby

(A)

'

., 6F

(B)

,

AF

J\x,v)=

ax

' '

f\r,v)=

d,

'"'

'("i=!+

AxAy

(o)

7@'Y)=tlr(x',)a'aY

6.

Two discrete

jointly

distributed random variable

x

and

y

are independent

if

(A)

pu=p"i

@)

pti=pi*xp*j

(c) Pii=Pn

(D)

Pr=l

(2)

7. If x=1and

y

=,

tt.o,r[I1l',

v

[z,v/

(A)1

(B)l

uv

(C)

v

(D)

u

8.

If mean:2,

standaxd deviation

is

Jz

for each

ofn

independent random variables

with

n:75

then by Central

limit

theorem Sn follows.

(A)

S,followsN(150.Jis0)

(B)

SnfollowsN(Ji50,150)

(C)

S"followsN(150,150)

(D)

SnfollowsN(Jt,

Jt)

9.

If

x(t)

represents

the

number

of

occurrence

of

a certain

event

is

(0,

t)

then the

discrete random process

{x(t)}

is called a

(A)

Renewal

process

(B)

Poisson process

(C)

Markov

process

(D)

Exponential process

10.

_{11 pr(') >}

₀

for

_same

n

and

for all

i

and

j

then every other state can be reached

from

every other state then the Markov chain is said to be

1A)

Ineducible

@)

Reducible

(C)

Transient

@)

Persistent

11.

If

S denotes the state space and

T

the parameter set

then

a random process

in which

T

is

discrete and S is continuous is called

(A)

Continuous

random

sequence

of (B)

Continuous random process second order

(C)

Discrete random

sequence

(D)

Continuous random sequence

12.

If

certain

probability distributions do

not

depend

on

t

then

the

random process

{x(t)}

is

called

(A)

Strongly stationary

process

(B)

Stationary process

(C)

Wide sense stationary

process

(D)

Markov

process

13.

R(r)

is maximum at

(A)

t:-1

(B)

r=1

(C)

r:0

(D\

r,:2

14.

A

stationary process has auto correlation tunction given

by

Rk)=:X#then

its mean value is

(A)

6

(B)

s

(c)

4

(D)

2

rs.

Rxy

(-z)=

(a)

-Rx,(r)

(B)

rx,(')

(c)

&x(r)

(o)

-&y(-r)

(3)

16.

If

{X(t)}

and

{Y(t)}

are independent WSS process

with

zero mean then the autocorrelation tunction

of

{z(t)}

where

z(r)=av(4y(r)t

($ onn(r)Rxk)

(B)

oznxx(r)Rw!)

(c)

..fono'(,)n,r(")

(o)

Jino(,)

17.

Real Sx.v(ro) and Svx(co) are

_

flrnctions of

o

(A)

Linear

(B)

Even

.

(C)

Odd

(D)

Complemented

18.

The mean square value of the process

{X(t)}

is

,

(A)

Rxx(r)

(B)

Rp1(-t)

(C)

-R>o<(-O)

1D)

Ru(O)

19.

If

the value of the output

Y(t)

at a time t-t1 depends

only

on

x(tr)

and not an any other value then the system is called

(A)

Casual

system

(B)

Time invariant system

(C)

Memoryless

system

(D)

Time depondent system

20.

The power spectral density of a random signal

with

autocorrelation function e-7t "1 is

(A)

llxz+c,tz

(B)

a/)"2+co2

(c)

q)" _/_{)"2 +}

-2

lol

u"t(M.,

+o;)

PART-B(5x4=20Marks)

Answer A-IYY

FIYE

_Questions

IfX

has the distribution

x

-1

0

I

2

p(x) _0.3 0.1 0.4 0.2

21.

22.

23.

Find

E(x),

E(

x'?),var(x)and

rar

(zx+t)

If

the

continuous

random variable

X

has

the

pdf

f*(x)=f,(r*r)in-r<-r<2and=0

elsewhere frnd the pdf ofy==x2.

The

following

table against the

joint probability

distribution of

X

and

Y.

Find the marginal

probability

functions

of

X

and firncti.on of

Y.

x

Y

1 2 7

1 0.1 0.1 0.2

1 0.2 0.3 0.1

24.

Show that the

random process

X(r) =

Acos(coot +

0)is

not

stationary.

If A

and

roo all

constants and 0 is

uniformly

distributed random variable in (0, n).

25.

Find the mean and variance

ofthe

stationary process

{X(t)}

whose autocorrelation

flrnction

is given

by

n(t)=to

+

-)

.

l+6r"

(4)

26.

The

power spectral density

fi.rnction

of

zero mean WSS

process

{X(t)}

is

given by

s(r)

=

{1

'

lot.l<

'o

.

Fird

R1.;.

[u,

olherw$e

27.

For a real random process

{X(t)},

show that Sr<(ro) is an even function.

PART-C(5x12=60Marks)

Answer

ALL

Questions

28.

a.i

For a continuous random variable X, the cdf is given by

[o

,f x<2

I

F(xl=1k(x-2)

2<x<6

I

[l

fx>6

Find (i)

the pdf

of X (ii)

the value of k

(iii)

Pfi>a)

(iv)

P(3<X<5).

ii

Find the moment generating function of the random variable

X,

whose

probability

frmction I

p(x)

=::

x =1,2,3,.... Hence

find

the mean and variance.

-2',

(oR)

b.i.

The

life

of certain

kind

of electronic device has a mean of 300hrs and standard deviation

of

25hrs.

Assuming

that

the lifetime

of

the

devices

follow

normal distribution. Find

the

probability that any

one

of

these devices

will

have

a

life

time

more than 350hrs.

What

percentage

will

have

life

time between 220 aad 260 hrs?

ii.

The frrst four

moments

of

a

distribution about

x = 4 are7,4,10,45.

Find

mean, variance Pz au;fi' _Po'

29.

a.i

lf X

and

Y

each

follow an

exponential

diskibution

with

parameters

one and

are independent,

find

the

pdfofU:

X-Y.

ii.

The

life

time

of

a certain

kind of

electric bulb may be considered as a random variable

with

mean

1200

hrs and

standard deviation

250

hrs.

Find the probability using Central

limit

theorem that the average

lifetime

of 60 bulbs exceeds 1250 hrs.

(oR)

b. The

joint

pdf

of a

two

dimensional

random

variable

(X,

Y) is

given

by

f(r,y)=ry'+*'l8

0<*<2.0<y<1.

Compute

P(x >l),P(y

<ll2),P(x >llY

<1/2

P(Y

<1 I 2 I

X

> 1),

P(X

<v)

and

P(X+I

<1).

30.

a.

Show

that the process

X(t)=

trs6t77

*

Bsin2t

where

A

and

B

are random variables is

wide

sense stationary

if

(i) E(A)

=

E(B)

= O

(1i) E(A2)

=

z(82

) and

(iii)

E(lB)

= 0.

b.

The

transition

probability

-"t

i*

of

u(fl?tov

chain

{)G}

rrl,2,3,....having

t}ree

states

r,2,3

is

,

=

[l.l

l.i

3.1'l^,,

the

initiar

disrribution

is

p(0)

=

(0.7

0.2

0.r).

[0.,

o.o

B)

Find

(i)

P(x,

=3)

(,,) P(x3

:2,

Xz=3,

Xr=3, xo=2).

(5)

31.

a.

Consider

two

random processes

0 is a

random

variable

x(t)

:

3

g.t1,

*

0)

and

y(t)

=

)sos(a

+ 0

-

r

/

2)

where

uniformly distributed

in (0,

2n).

Prove

that

artr

(0)

ftry

(o) 2

n r

(r)1.

(oR)

b.i.

The

ooss

power

spectrum

of

a

real random

_Process

{X(t)}

and

{Y(t)} is

given

by

s*

{ar) =

[a

+

jba'

lol<l

. Find the cross correlation function.

^r\ ' |.0

-

elsewhere

ii.

lf

y(t):

X(t

+

a)-

X(t

-

a)

where

x(t) is a

wSS

process

then

show

that

Rn(r)

-zno(r)-

n,o(r

-za)-

noQ

+za)

32.

a.

A

wide

sense stationary process

X(t)

is the input

to

a linear system

with

impulse response

h(t)=2"-t',,rr

0.

If

the autocorrelation function

of

X(t)

is R>o<(t)

=

e+l'].

Find

the power spectral density of the output process

Y(t).

(oR)

b.i.

Find the power

spectral density

of the

random process

if

its

autocorrelation

function

is

given

by

Rxs(r)=

s-"F1 sss

3r.

ii.

The

short

time moving

average

of

a

process

{X(t)}

is

defined as

Y(i=+i

,(r)*.

t-T

Prove

that

X(t)

and

Y(t)

are related

by

means

of

convolution type integral. Find unit

impulse response

ofthe

system also.