7/25/2019 ass10_soln_0

1/23

MSO 201a: Probability and Statistics

2015-2016-II SemesterAssignment-X

A. Illustrative Discussion Problems

1. Let random vector (X, Y) have the joint p.m.f.

fX,Y(x, y) =P(X=x, Y =y) =(x+y+k 1)!

x!y!(k 1)! x+y(1 2)k, x, y= 0, 1, 2, . . . ,

wherek 1 is an integer and 0< < 12 (we say that (X, Y) has bivariate negativebinomial distribution). Find the marginal p.m.f. ofX, the conditional distribution

ofY givenX=x (x {0, 1, . . .}) and the p.m.f. ofZ=X+ Y.

2. Let the bivariate r.v. (X, Y) have joint p.d.f.

fX,Y(x, y) = (1+2+3)

(1)(2)(3)x11y21(1 x y)31, x >0, y >0, x+y 0, i= 1, 2, 3 (we say that (X, Y) has bivariate beta distribution). Find

the marginal p.d.f. ofX, the conditional p.d.f. ofX given Y = y (y (0, 1)) andthe p.d.f. ofZ=X+ Y.

3. Let X1, . . . , X n be i.i.d. A.C. r.v.s with common p.d.f. f(

) and common d.f. F(

).

For r {1, . . . , n}, let Xr:n = r-th smallest ofX1, . . . , X n (so that X1:n X2:n Xn:n are order statistics of random sample X1, . . . , X n).(a) Find the joint d.f. of r.v. (X1:n, . . . , X n:n) and hence show that it is A.C. with

a joint p.d.f.

g(y1, . . . , yn) =

n!n

i=1 f(yi), ify1< y2< < yn0, otherwise

.

(b) (Relationship between beta and binomial distribution) Let X

Bin(n, p),

wherep (0, 1) andn N. Using integration by parts, show that

P({X r}) =ni=r

n

i

pi(1p)ni = 1

(r, n r+ 1) p0

tr1(1t)nrdt, r N, r n.

Hence show that ifX Bin(n, p), thenP(X r) nr

pr.

(c) Using the fact that, for any real number x, the event{Xr:n x} occurs iff atleast r among X1, . . . , X n are x, find the d.f. ofXr:n, r = 1, . . . , n. Using the

1

7/25/2019 ass10_soln_0

2/23

relation derived in (b) show that Xr:n is A.C. with p.d.f.

gr(x) = n!

(r 1)!(n r)! [F(x)]r1[1 F(x)]nrf(x), < x < .

4. (a) (Poisson approximation to binomial distribution) Show that limnnrprn(1

pn)nr = er/r!, r = 0, 1, 2, . . ., provided limn npn = (i.e., for large n

and small p, so that np = > 0, Bin(n, p) probabilities can be approximated

by Poisson() probabilities);

(b) The probability of hitting a target is 0.001 for each shot. Find the approximate

probability of hitting a target at least twice in 5000 shots.

5. Let Xbe a non-negative and integer valued r.v., satisfying P(X= 0) =p(0, 1)and having lack of memory property P(X n +m|X n) =P(X m), n, m{1, 2, . . . , }. Show that X Geometric(p).

6. (a) (Binomial approximation to hypergeometric distribution) Show that a

hypergeometric distributed can be approximated by a Bin(n, p) distribution pro-

videdNis large (N ),a is large (a ) anda/N p, asN ;(b) Out of 120 applicants for a job, only 80 are qualified for the job. Out of 120

applicants, five are selected at random for the interview. Find the probability that

at least two selected applicants will be qualified for the job. Using Problem 6 (a)

find an approximate value of this probability.

7. (a) If the m.g.f. of a r.v. X is (1/3 + 2et/3)5, < t < , find P(X {2, 3});(b) If the m.g.f. of a r.v. X ise4(e

t1), < t < , find P(E(X) 2

Var(X)0.

13. (a) Let F() be the d.f. of the r.v. X, where P(X= 1) =p = 1 P(X= 0). Findthe distribution ofY =F(X). IsY U(0, 1)?(b) Let the r.v. X have the p.d.f. f(x) = 6x(1 x), if 0x1, = 0, otherwise.Show that Y =X2(3 2X) U(0, 1).

14. (a) IfX U(0, 1), show that ofY = ln X Exp(), where >0;(b) Let X1 and X2 be independent N(0, 1) r.v.s and let Y =

X1X2

. Show that Y has

the Cauchy distribution. What is the distribution ofY =X1?

(c) Let X and Y be i.i.d. r.v.s. with common p.d.f. f(x) = c1+x4 , < x 0, show that (x1 x3)(x)< 1 (x)< x1(x) (Hint: Use integrationby parts in (2)

12 (1 (x)) =

x

t1(tet2

2)dt).

17. Let Xbe a r.v. withE(Xm) =m!, m= 1, 2, . . ., and suppose that the m.g.f. ofXexists in a neighborhood of zero. Find the p.d.f./p.m.f. ofX.

18. (a) Let X Gamma(1, ) and Y Gamma(2, ) be independent. Show thatU=X+Y andV = X

X+Yare independently distributed; also find their distributions.

(b) Let W Beta(1, 2) and T Gamma(1+2, ) be independent. Using (a),show that W T Gamma(1, ).

19. Let X N(0, 1) andY N(0, 1) be independent. Find p.d.f.s of X|Y| and |X||Y| .

3

7/25/2019 ass10_soln_0

4/23

20. Let X1, . . . X n denote a random sample from the Exp(1) distribution. Find the

marginal distributions ofY1, . . . , Y n, where

Yi =

ij=1 Xj

i+1j=1 Xj, i= 1, . . . , n 1, Yn= X1+. . .+Xn.

AreY1, . . . , Y n independent?

21. (a) Let X1, . . . , X n (n 3) be a random sample from N(, 2) distribution. FindEXS

.

(b) Let X1 and X2 be i.i.d. N(0, 1) r.v.s. Find the distribution ofZ= X1+X2|X1X2| .

22. LetX(1), X(2), . . . , X (n)be the set of order statistics associated with a random sample

of size n ( 2) from the Exp(1) distribution.

(a) LetZ1 = nX(1),Zi = (ni+1)(X(i)X(i1)),i= 2, . . . , n. Show thatZ1, . . . , Z nare i.i.d. Exp(1) r.v.s.(b) Using (a) or otherwise, find E(X(r)), Var(X(r)) and Cov(X(r), X(s)),r < s.

(c) Show that X(r) andX(s) X(r) are independent for any s > r;(d) Find the p.d.f. ofX(r+1) X(r);

23. Let X1, . . . , X n be i.i.d. nonnegative A.C. r.v.s. with common d.f. F(). If E(|X1|)< and Mn= max(X1, . . . , X n), show that

E(Mn) =E(Mn1) +0 (F(x))

n1

(1 F(x)) dx.

24. Let X and Y be independent and identically distributed N(0, 2) r.v.s. Let U =

aX+bY and V =bX aY (a = 0, b = 0).(a) Show that U andV are normally distributed independent random variables;

(b) Show that X+Y2

and XY2

are independent N(0, 2) random variables.

25. Let X be a random variable and let and be real numbers such that < <

7/25/2019 ass10_soln_0

5/23

B. Practice Problems

1. Let X= (X1, X2) have the joint p.m.f.

fX(x1, x2) = x1x236

, ifx1 = 1, 2, 3, x2= 1, 2, 3

0, otherwise

,

and let Y =X1+X22

. Find the p.m.f. ofY.

2. (a) Let X Bin(n1, p) and Y Bin(n2, p) be independent r.v.s. For t {0, 1, . . . ,min(n1, n2)}, find the conditional distribution and conditional mean of X givenX+Y =t.

(b) Let X Poisson(1) and Y Poisson(2) be independent r.v.s. For t{0, 1, . . . , }, find the conditional distribution and conditional mean ofX givenX+Y =t.

3. Suppose that the number, X, of eggs laid by a bird has the Poisson() distribution,

and the probability that an egg would finally develop is p (0, 1); here >0. Underthe assumption of independence of development of eggs, show that the number, Y,

of eggs surviving has the Poisson(p) distribution. Find the conditional distribution

ofX givenY =y.

4. (a) IfX Poisson(), find E((2 +X)1);(b) IfX Geometric(p) andr is a positive integer, find E(min(X, r)).

5. Let X U(0, ), where is a positive integer. LetY = X [X], where [x] is thelargest integer x. Show that Y U(0, 1).

6. IfX U(0, ), where > 0, find the distribution ofY = min(X,/2). CalculateP(/4< Y < /2).

7. LetX N(0, 1) and letY =X, if | X| 1, = X, if |X| >1. Find the distributionofY.

8. LetXandY be i.i.d. U(

1, 1) r.v.s. Find the probability that x2+Xx+Y = 0>0,

for every real number x.

9. Let X andYbe respective arrival times of two friends A and B who agree to meet

at a spot and wait for the other only for t minutes. Suppose that XandY are i.i.d.

Exp(). Show that probability ofA and B meeting each other is 1 et.

10. LetXandY be independent r.v.s. with respective p.d.f.s. fX(x) = 13

, if 1 x 4,= 0, otherwise, and fY(y) =e

(y2), ify 2 and = 0, otherwise. Find the d.f. ofT = X

Y and hence find the p.d.f. ofT.

5

7/25/2019 ass10_soln_0

6/23

11. Let X andY be i.i.d. U(0, 1) r.v.s. Find the marginal p.d.f.s of

(a) X+Y, X Y, X+YXY,|X Y|;

(b) min(X, Y), max(X, Y), min(X,Y)max(X,Y) ;

(c)X2 +Y2.

12. LetXandY be i.i.d. N(0, 1) random variables. Define the random variables R and

byX=R cos,Y =R sin.

(a) Show that R and are independent with R2

2 Exp(1) and U(0, 2).

(b) Show that X2 +Y2 and XY

are independently distributed.

(c) Show that sin and sin 2 are identically distributed and hence find the pdf of

T = XYX2+Y2

.

(d) Find the distribution ofU= 3X2YY3

X2+Y2 .

(e) LetU1 andU2 be i.i.d. U(0, 1) r.v.s. Then show that X1=2 ln U1cos(2U2)

and X2 =

2 ln U1sin(2U2) are i.i.d. N(0, 1) r.v.s. (This is known as the Box-

Muller transformation).

13. Let X1, X2, X3 be i.i.d. Gamma(m, 1) random variables. LetZ1 = X1+X2+X3,

Z2= X2

X1+X2+X3andZ3=

X3X1+X2+X3

.

(a) Show that Z1 and (Z2, Z3) are independent and find marginal p.d.f.s ofZ1, Z2

andZ3.

(b) FindE(Z21Z2Z3).

14. LetX(1) < X(2) <

< X(n)be the order statistics associated with a random sample

of size n ( 2) from the U(0, 1) distribution. Let Yi = X(i)X(i+1) , i = 1, . . . , n 1,and Yn = X(n). Show that Y1, . . . , Y n are independent and find the p.d.f. of Yi,

i= 1, . . . , n.

6

7/25/2019 ass10_soln_0

7/23

l'1So

Zot

c,.

',

fvor5all

f.t+3

ct^J,.

S+Jr

A*r

c

/1

otS-

:.o\6

-

It

Ser*eA*ev

Nnni?h\.^en{-.X

SoLq4io

r,.z\

t^.\

Ard

.1\-

(-t

1;

,L["]-')

^'-

'

-

t

anA

b-,

}sx

rr-rs)* ;1\s)K

b.\D

I

)t(k/

\"-

t--

tl'rK

\_Lor.

'l

o

'

ot4zv"rtnc'

L

I

U:

0^.tL-t)

z X 0"J 2t

{^tt '1

l?-

y'+\.

-{f,"

=

{

(t-:\

ert

,

l>g

);'-

1*1

7/25/2019 ass10_soln_0

8/23

AhA

**

Notxrr

\'A'lo.

sl

lu-

at

o'

[u-r\:r),

\**,[1

r*t

\5\

' ,r,]"=

{r*}t*t1*rF]

t o

'*:;"':''

CIe-r.A1,

In'v

\^

I

Fov

urelor)

\r\")

:

\o1,^1

'

$ytSaa$1

t.'.

q

_(

-b

A1

=

\

s*-qs,

,

^$'-,

i

sr\r\

;

\u

rlfrt

Clo-"ll

tr-,

G"

ilu,

\.v

)

t3\)-

Sr.-1

1

t\-3)

-

C

,^

A

b

,

\ot

Jc(or).

/ \ rsr-t

\;

(t+)

t,-ir)

,

r\ b4).

7/25/2019 ass10_soln_0

9/23

lC

fotx*

a\q

-,-

)^\

>

A,\.

{

(Xr:*zX}:h,,--2X^.,.}

;\

\tY,,r1)r

X1.,^1)r,. -,,

Xn,.

1)-)

kl

\

/r,.^

1)r

Iu

n

L1L_.-._

X,^,-

9).,

XD,.

-Dr.iil#tl]

t

t

t

xg,

1l

YPrtTr.-''

t

/.y^L)n,

xP,'

rgr4'-'

4xL)

]

csr^

L

|

[*r

t

1'-

Y1

3)u-

''

'

z

*"'

9

J"

-t,..,].:,

-;.;*^'

eS.

Lr-

t\;,

r:;

lr- 1rr.-'-

*^

):

xrh\"=r)

--

I

tx?".1

Pt..x;r)

>

L\-h)

ltr;")rn=y\'

'

L

l\x7

r): Lr-h

)

,

R:

(r-Ltt'z\io'e

PIx;r")t

tt-lt)hv"4ttt

x)^\

-{tx)n+r

):

}ttr-Vl^-

h z\

r-

"

[\rtr+l)

i\

\>s-t-LJ'''

4

X^rc.c.,^et'rcL]).

1 o

3-h''l^'$r/'e

t2

v" [q,-

N".t

La)

u^

y^rtrl\\t:ur.

T.

X*\1o"tc(y'a

A'['-.cr**'

r,raL[

o,

\-rr.'(z

"

\z

lr>,

a1\

a

\^tr

)>

g'

l,:*',1\)

t

r-H\\)"

ffil

f')

)

\\-ra,,

fr)

\

K f,

\

{

I=l'-',rt:

tX)

kt,-bl-''

.

(

LLl

V-t\-o)'-:

\

k>

e

\\'"

"'

rtt"tt

\ao''^

'

ol,,r_.ar^r_

z.2\D''

Tt-r

[t.^

lr\-r

[5)

L4Jr

'/;

+

4

a\\,0ta*rA

i

O

.1yQ_

"1pt

e.l.\t

e-t

X

^,

tta\

1

go_

{.

I

r-o;

r\

-'|t.tr

Ac.le,'J'.-r

A'-

le

4J

f,""

&

d-t

.

Rept'.-r-

\n"ra5r ,t'*)

=

ltx)

u)

>

l--

tt.x>t)

-\

k>t

1

x

"\R/

I)r...

\

S,

t

)

.

k\ *,'.**

\=a{:?,}::i:ii?r.

I

>

\-

(l)

\bf

rt)'-

tE

\t

lv

)rt)'

7/25/2019 ass10_soln_0

12/23

voblt..

U".?

\A)

ctaug'1

f,

nu

br"

tt

3

)

l)\xc1>-s\

)

;

tt")

6)"

Gl'-tl)

G)'(t)'=

+

X

".

totr'"*,r

tt\.

1G,,,

ELrl>

Vav(x

l>

r.

Jffil

(x

(Etxlr-r[il*.,),

f

\-")

rV.

t \.'

l>r

l-L

^.K

I

Y-\

\{

-l

rl

-e^

q*

b\

o,

1

;-

g)""

iot)'\:

E.+

- tx r)'

pr

l-).

Jzb

L-I

S..\\orrt-

4^*^

lrt-r

\'.*i

15

O:

^.

ttt*".t-

ax'l'

X

=

+

4

tu

Jn

't'^

'l'*-'

\r"^'

h

'

-IL

q

).

^,

b*

L

t',

\,)

'

t(

-\r."

\c"^*

1

Vse

va\ft'e

l-,

c\

I-c./\'\

a U"l

Ax\

4

tt')l;

ll l:

\

vr-r)

z

t-1\{zo)-

it*z

n)>

\-

HT-

-

h-f

=

\- 4_.

),

o.15

+)

f

>

qo b)

h?

[

vr-x

>\

)

\-l

f-\

LlC,^

N",

\o

corr6rrqv

^

[e1y,,c,^c.

4

:^^;\"".t"1,

^]n"l;:i:t

f;"tryt^

ro\t^

\t

i

+:

"\

/ru"tx

caA

tn

{

t

t\\

1*

t,-\ )*",

h:v

t

Au"cce.nn

(nn

\v*b)4t

\vt't

)

F=o

i

t"T:l

)$'"\v )

h

\>o

---1r-/

(g^t

)

n\'\((

7/25/2019 ass10_soln_0

13/23

)*".*

n*.a:

L.*

1:

\+.

Lte rrcaa\

&

nus'-D

--|uJ

Ltl',

1"

t'-\'^-n

=

\"

F-

[^:')

t'-r:k

e) i

[1.)

L,-r,)*-"

on-"-

i"

f"

I-')o*.

l2v \c=-

L\rS:

\.\r.,....r*

tt

A.

7/25/2019 ass10_soln_0

14/23

?vo I.c.-

\\s.

\L

he t^r{

A

uj\a

-tr.t

e

S4r..

I.t,'1

a- \

v"ri'^Jto.

.*

?\

t.e,

l5

*

\+{

h

-}J,

l-r.vp

,-r

\^

.lrr

d

^-,

o {;ir r^ ^**

2

T*

Lns:

Uj

k-t

,

\:

t

r-l'

)k

\JAg

c

.[e-."

[1

7

G

e-Lu*"-"jn-^i4

tt-+f

Cf

[*I-,r

lJ(

6t

.-

t"-'l

1v-t

1"

-C

dK^*

r

,-

4i

lJr

E)'"-i

___--

v v

,

t\_+)

v

\)c

:

+

{^

+

t\*bn*\

)

Fl'*\

u\'h^

'

-'/

\lv'\"stli+;

Ota

Cr.)'

1

ca

tvo tc.-

b\o'

\)

vol

7/25/2019 ass10_soln_0

15/23

(b

J'to

O.e.

.')

ie;

o2

t^tt"-rt):

Io_").

T'*

-\l*s

.d,\,"\

].^

\t'ul=

lg^

1d\rr)

\

hritrr

\

1.e" ')

=

*\di)

\

-\u']\ te.)'\=

\

1

$)

-[r*

forr^r

\

*.',.

d

{-*L

^

L

r,

,J

1t;":'-

:

+

i

'ry',

or*.t*t

sxy*'

lL

C.*i.r-eu

ln"r

\-

\

*vot"n

\o"

t-

'irt..-r

tIl

Xil

--

\"-

*-

,

z='/'u

L.e.

X1

z

-1

?, /"?Z

SJ.

=\t'l,rlt

-01

'7L

7/25/2019 ass10_soln_0

16/23

Ur**tr

L

\-

"-t'

\i6

L\-

6\r\

)

Nt'n'

[r^

t\-

$\rt

)

-

-

*/,

z

\

)=

\t

q:

-u

?

l1t

e-*7'rt-r)

-

i

P

->'A

-r-L

1;

L?e1

t

l

'

'o:

)

l

t

'(?l

q.c

)

L

t+

e-t

)

a+

--'v-

F

*,

.-....-

)\

\u,\*'\

:

\.rr

.r,

)

d.qr

\rl.

u,

4

J

D

q

I

e

; >ct

9r(

t

\-"1

\

r

_,^i

\.I..,

\-

ot"l

*"1

t

\-t

u|'rl

o,

c"nrt

x)-

S

g^

'1-u

ut

) \.\)'i

\x?1'6

)

:

\

1 c>

-

I

\

\4

r'tr

bqt*.)du,zr

)

-

e1

o)

:

\-

9

[>{

1:

\-slr)

Lb)

l\1'6

I

x

1\5'El:

o'4os'

-',.:^l*-tlltJ;tr-t

\x>'1'e))=o'+s"r

I \

()

1-t)

>'tr51

-)

\-

g\^:;

):'8\s1:4

g

(-t#

):'ersl

o.\

-)

}a-O'1t-?

1'("'-

(S)

tq)

\

.

?o'l

?'\rtt'

2 l-

=EI-SI

a rL

J

e-t

u^

,r

L,

-*

/t

Lrse

4

rI

)r

*t-

\r

te. V^

%n^

t-

r'e

_\

r

)-

I

-

L.-

e,

-. t

-

e-

-+(.1

e'**

I>

'i

t"

fv- lq*

t-\s.

\5

7/25/2019 ass10_soln_0

17/23

l"' [.,-

\o.

\?

q

)

C.rnnril.cv

-h^s

ts\

{vat^n

\ou'*At"-

(x,r)-)

U:-

**4,

V

z

*-,

-\

x"

U$,

-l:

u

Lr-v)'

Su-v"

[\uv):

\^\]>o-

t\(\'t]))"t1

:

{\t"l'

:

\o_")*

nx\r\:

i{utxi)=

i*i'

)z>

rr

)2>

,-

gv

\

(r

)

d

x

..4

N.

c

.

\,)i"\t^

\.

4,\'

,to*.\L1)

>

dA

^+::'"-

\

\

= -r

\

l&l

U'

iTr

,

IyJ}-l

t

u/"F

J

'^

ci5"J"

r"'c*t""t'

J

7/25/2019 ass10_soln_0

18/23

r rr r h,.,

v"ble.-

No,

2-o

Co,,.r..r

J,cv

ld^r

\-\

Jkvr^r^n

\orr^

,.Jrco'l

t

:rr .

.,,

*n)

-i

L\r

:

"

I,^)

d.Jot'a.\

5f

h{

\.

-

A

,

lr

r

*r

7/25/2019 ass10_soln_0

19/23

=)

\1

..

.

a.

Avc

t?).

'zhzt

6^-\

tt.blo,*.

tF,L\

\

*,

\,.:c

f

P

st\1-')

E\t)'

E\1',t

-L

t\J :-'.,

X^-t

=\

g-L

#tr

-\

O,

.) .

(^rq

(r^

Aelc.^^Jre^*

gr^

Ac

\

or^t

er"

l

\1^\r

\

=

t

\

Lnorp

J\

\*

rll

t\

\-

vr

-6q

E\4

)'

ELt)'

\ h-L

\5-

Fl

tb

)

L.*

\r

2

/rt

Yr-

6.^r.

ar

"-

Tar.r,[*\

{"\;

E

\etr\r-t

\r'av

z)

fl-r-r"ttY+'\

=

J.\

\r

\

-l

4

.\L

={ \1

\r-

n-r-

1;

Cr.vc

Lwtt-\

r-r^Aoat

*1

l

\r-

L

6^^f \

)-

-\1

a\rc

q

r^ d'

e

\

c'^"\

Ct

\

y'1I

\

\^.Ic\r^lcr^+

.-7- /

l-

C-

*1-

{r-.

1i-'

)

--

\

es**t\

jt

o

L*r{r'\*>;

t

9t

a^-*.:l.u

tnJ.c\c,.^lcr^1

|

\

d.,\.

bv

Ca.*c\7

ol.t"x

l

7/25/2019 ass10_soln_0

20/23

ffil

fo*[>tr

fr,f

r,*.

1

"

d

F--

,

\r.s1

=

\o-',)n.

by

lvolle'*^),

)ots\

\'*'t'

d

t{crr'-''

/1*t)

n

l

]

t\r:.-"1.)

?

t

\5

cLlt

'

{'

oanc"'

"^'o:

\

o

,

#orrAo'

LorArl.p^

-h{

\-\

-lqvatnn\ov,,^.Jr.'o-

L*t*)-",

Ag^t)+

lz:

"'-

aul

Ar\t*.-t

\1

?t"

h*cr'\

?rzt^-\\\*ttr-ltrr\

2i'-

\rra'-ti\)'

I

Xrtt-*11-tr

\r

"

5'

"

''\'

tj'?

''

ytr\

?

n+

,

xrr\

?

?*

-{

?-4,,

>(ttl?

?r*

'*f lr'

'

'

',

{1^1

:

v)

n

vyn-r*

"'

S?,

r

L,

o

z-

+,

.

L+-e

s

h*

}r*-t

t*,

-'"

:t*s#"'*\'*1t

:

r-\Lt

tr?o-

\>7o'

"'z

T.)"J

\

r^

o

D

os

o

\

\ , r\

oo

o

\ f

51

\l*i."*.,

"oo

o

\?

t

\

\r^l

lL.L\\

\ . \

..

.

*i

\.u+-h"

l'a^

lf-r

-t)'

+ r>\

'42t1

):

t

D-

e

t?r

"i

(\

"bt)o,

u=L

"'-'

D. l'

Evy

{\)

u

-v'

b'J I

ur^\n7r

t'i

,.i.o*J'

\,,^

Lo'

t

)

Au'sr"

-\^'7\

t^h

4'

\

4*-\

\>'^

U

)*'

\-\.,ttc-

'V^-^

C^.4

A*J'

-\4*^-dsra

C.t'Ut

a

(tF\

1

C'\r "^\-

U-

t^>

\

-\

\^n

ht\

\

t

6'(tut

I

sn

4

(\\

v-h)

\

>\)rLct\t-c\t'-\t)-aL\)[oh4