MSO201A/trSO 2O9: ProbabilitY and2OL2-2OL3: II Semester

Quiz IITirne Aliowed: 30 Minutes

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Maximum Marks: 20

Name:

Roll No.:

1. Let X be a random variabie with moment generating function1 i , 11 ,Mx(t): G + :u-i + iu',, R.

(i) Find P(2x2 -

x -

r:0);(ii) Calculate E(X);(iii) Calculate E(X2).

2. Let Z -

N (0,1) and let h : R -+ IR be definecl by h(r) : max(r, 0), 2 e R.. Find theEo M*dlvariance of h.(Z).

56L"'- t (a) |

iE *l*?n(

U-,*tI

--Lcqv , ttt- 7I

e", LgrR

3 rlar'Lua6r*, p(zxL-x-t=o)

= ?(Ox+l(x-rl =o)

m&! 4 ^fanJrn Vtu>iab)2. X ol.o"scis ff*" bJ

?(x=-+ d x=t)-7(x---+)+?(x=t)I , L7

_

-,1- _Lt i"6EL6

2 HarK;

L na^K

x o LL 1-{*k) L16 L\ -LL6

0x+ -+tL + txL16 Z' L, ' G' TcI , llI tL

Ll(

(6) Lg) =

Mh".a (x)

rlJ t=o

l* ",t*f r=of t -Ll,rt d''' *+ e! + Jl-B l(

Ll6

o"x f^ + t+)'L+LIL I C3T

JIIC

xLt1

f*FTxrtlo=,[+ "\ + +i .*] L=o

E (x') =(c) tlr'x

AIkr,E (x")

=

1, , ll-

T[( I(3LT

2 tlaat.-,

So{.": 2:

E (uL'))Vott (trt )) = E(hb))'_= t (yn^x (=,0 ))

= i*o, (b,o) t 7"/,-a Jzr(* z ev"l-Ag=\'o -EA

4t(LJfi Jo-

I

r--J zx

(e thb)) )"

^>

C (he))' = =(rnor

(z,o) )'= \U^x(z,o))" h r*/'rb

-6"O

- f -t-"

-1.-)6^D

I

GI

-

--.

JfrL

qL

-bl-,e. '- d>

I]*"' e L 'LL

2 tlr*rtli">

2nanl6

7 narr K;

L lMo.rt{>

lsl-lz

L

T-IzT

2 Ylanka

Var'(nt,)) = r r 12\c^l