SMCS

Statistical Models for Computer Science Paper: CSE-405 UNIT-I 1

(a) (a)

Desc Descri ribe be the the pos possi sibl blee samp sample le spa space ce if if an “if “if … then then … els else” e” sta state teme ment nt is is executed four times.

(b)

(10)

Use the axioms of algebra of events to prove the folloing

relations !

!

(i)

"U "#"

(ii)

"U $#$

(iii)

"% &#&

(c)

Defi Define ne the indep ndepen ende denc ncee of of eve event ntss and and also also the ter term el elia iabi billit. t.

(a) (a)

+hat +hat do do ou ou mea mean n b ,ro ,roba babi bili lit t mod model el - +hat +hat is is $am $ampl plee spac spacee - $ta $tate te and prove aes/ rule.

(b) "

(')

+rite don all axioms of algebra of events.

(*)

(1) ()

2n manufacturing a certain component3 to tpes of defects are li4el to occur  ith respective probabilities probabilities 0.05 and 0.1. +hat is the probabilit probabilit that a randoml chosen componenti) Does no not have bo both 4i 4inds of of de defectsii) 2s defectiveiii) iii) 6as 6as onl onl  one one 4ind 4ind of defe defect ct give given n tha thatt it it is is fou found nd to be defe defect ctiv ivee- (10) (10)

" series series of n78obs arrive at a computing center ith n processors. "ssume that each of the nn possible assignment vectors (processors for 8ob13……3  processor for 8ob n) is e9uall li4el. li4el. :ind the probabilit that exactl one processor ill be idle. (10) 5 (a) 2f a1 and a are an to events3 prove that ,(a1 ua) #p(a1);p(a)7p(a1%a). da). (ii) "t least to person have the same birthda.(month>da) birthda.(month>da)

(10)

# (") Derive Derive the product la of reliabilities reliabilities for series series sstem sstem and product las of unreliabilit/s for parallel sstem. (10) (b) consider four computer firms "33?3D bidding for a certain contract. " surve of past bidding success of these firms firms on similar contracts contracts shos the folloing probabilities of inning !

 

,(")#0.@53p()#0.153p(c)#0.@3p(d)#0. efore the decision is made to aard the contract3 firm  ithdras its bid. :ind the ne probabilities of inning for "3 ? AD. (10) B. +hat is ae/s ule- =he source of incoming 8obs at a universit computation centre is 15C from du4e3 @5C from orth ?arolina and 50C from orth ?arolina $tate. $uppose that the probabilities that a 8ob is initiated from these universities re9uire set7up are 0.013 0.05 and 0.0 respectivel. :ind the probabilit that a 8ob is chosen at random is a non set7up 8ob. "lso find the probabilit that the randoml chosen 8ob comes from the Universit of orth ?arolina given that it is a non set7up 8ob. (10) . Define ernoulli trial and generaliEed ernoulli trial. ?ompute reliabilities of m7Fut7f7 n $stem as a function of reliabilit  using n# @ and m#133@. (10) UNIT-II

(a)

1

Define ,G:. $ho that the sum of to independent random variables is the product of their ,G:s.

(b)

(10)

2? chips are purchased in lab. Het I and J denote the times to failure of  these chips purchased from to different suppliers. =he 8oint pdf of I and J is estimated b f (x3 ) # K 1 K e

7( K1x; K)

 here 0 L x L M and 0 L  L M.

:ind the marginal densities of I and J. :ind the 8oint distribution. (10) !

(a)

+hat if N==:- Give N==: for an to distributions.

(b)

Het I and J be to random varibables. =hen prove that

(c)

(*)

(i)

2f O # I ; J3 then N>m 9ueuing sstem. (15)

C(ITE(I) *+( INTE(N), )SSESSMENT 2nternal "ssessment! 5 Nar4s of "tt.! B.5 Nar4s of "ssignment! B.5 Nar4s of $essional! 10