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MSO

201a:

Probability

and

Statistics

2OL5-20I6-II

Semester

Assignment-VIII

A. Illustrative

Discussion Problems

1. Let

X:

(Xr,

Xz,

Xs)'

be a

random

vector

with

p.m.f.

f

xr,xr,xr(rr,rr,rr)

:

{

i'

if

(21'

12'rs)

A

|.

0.

otherwise

where

A:

{(7,0,0),

(0,

1,0), (0,0,

1), (1,

i, 1)}

(a)

Are

Xt,Xz,X3

independent?

(b)

Are

Xt,Xz,X3

pairwise

independent?

(c)

Are

Xt

-l

Xz

and

&

independent?

2.

Let

the

r.v.

X:

(Xt,X2)'have

the

joint

p.m.f.

(b)

Find

P(Xr+Xz)2),

P(X?X2>3),

p(Xr:2lXz:1)

and

p(Xt*Xz>

2lXr:

11.

3. Consider

the following

joint

p.m.f.

of r.v. (X,Y):

(

rt*2rz

f

5@r'rz)

:

I

-it-'

tf

r1:

r'2'

12:1'2

(a)

Determine

the

conditional

p.m.f.

of X2

given

Xt:

rr, Tt:

I,2.

f

x,v(r,Y)

aI

r-+

1

2

3

4

4

5

o

08

04

09

11

12

06

09

27

OB

03

05

04

(a)

Find

the conditional

p.m.f.

of

X

given

Y

:

b.

(b)

Find

the

probabilities

p(x

+

y

7),

p(xy

18), P(X

:3V:5)

and

P(Y

:5lX:3).

4.

The

joint

p.d.f.

of

(X,

Y) is given

by

(

.,

fou(r.r.,)

:

J

c(r

-

r

-g),

if

r

> o,a

) o,r*y

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(a)

Find

the value

of

the

constant

c;

(b)

Find

the marginal

p.d.f.s

of X

and

Y;

(c)

Find

the conditional

d.f"

of

Y

given

X

:

r, where

r

e

(0,1)

is

a fixecl

constalt;

(d) Find

P({2X

+

3}'

o

ilii,

;?

e

e

-,

)

:

\-:.:::i:::::'"

1

1x3:')

-$

i

-

o*J

X,, a\rq \n-'r '

..'^ {o

\

t\

\\-l

Xr

i

>

Ao

-\

w'

r')

Ao;r^J,

Sr={o'

La)

u

\;-

Xr-tr>

(J-

'\

)"o-)

\r\r\'

\tr;-l)

7

\t'

\.*li"-t'^'

(

L 4

r,'r

r)?\o-t)-

\t-\)

\ \

)

i1t':

(ro)

\-',n)t't)'\

Lr'

.\^

t

';

.

o4rrvu'tnr

\\'o,

rr>\)-t

+ \\t")l\xl=\)-- t*ltd'

2\

-\>-

*1

o'\re

t^"\

tr^\c-\

'

r"l'gr'"h'

*,]]e^r9\*'.'")=

?

'

\*rz>)>

\''l\x"-r\:

'"'\

\xa2,)t

+

?j.,l;:;,i'

\

\

x,-r'r

\*1-

\)

:

r

ttr

^'*1''

\f-'l

T:

\

xu\x.,\r'-\>)

'-

t\x"r-\

/':-2'

=

t

r*'"i

rl:i'i;)

\'J

ofi7a.tn

;

\

:;

..\

o^",',

'

\

';

s.\r"raera*

ltb)

?[y,\

\g

\Y

^,.t"

_

\,

.\ ,r1ir"'LrX1z\)_,ut

\

tf

/r-)3):

?

(xrt

>\

.\'.

\xt'>\rg't\=

t

''

[(

or;'--'i

V\*r-r

*'?t\ X>:-r);

P

\-*r)

\\I1z\)

:

\

vol

0r..^

.

b

lt"l

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=

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