Download DU (University of Delhi) B-Tech (Bachelor of Technology) 5th Semester 6191 Probability Theory and Statistical Computing Question Paper
Sl. No. of Ques. Paper : 6191 F-S
Unique Paper Code : 2341501
Name of Paper : Probability Theory and Statistical Computing
mee of Course : B.Tech. Computer Science
semester : V
Duration : 3 hours
Maximum Marks ' ' : 75
(Write your Roll No. on (he lop immediately on Icccipt of this qucniou paper. )
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Nom? Answers may? be written either in English or in Hindi; but the same medium -?
should b? used throughout the paper.
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QuestioQ No 1 is compulsory and 15 of 35 marks (7X5? =35 marks).
Attempt anyfaur questions from Q. No. 2 to 7. (2 X5 10 marks each)
' ? Bdrm of a question must begnsweredtagether. .
Use of Non~Programmable Scienti?c Calculath ls allowed.
The symbols have their usual meaning.
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2(a) State Baye? s theorem. Stores A, B, and C have 50,75 and 100 employees and respectively
50, 6.0, and 70 percent ofthes?e are women. Re?signa?ttons are equally likely among all employees,
megardless of Sex. One employee resigns and this is -?a? woman. What is thQ probability that she
1 works in state C?
' (b) Suppose that an auplatte qngine will fail, when in ?ight, with probabllity 1 -- p independently
? ?'pm; engine to engine. Suppose that the airplahe will make a Successful ?ight if at least 50
percent of its engines remain operative For what values of p. is a four-cnginc plane preferable to
a two-engine plane?
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2
(e) Let the probability density of X be given by
_ {c(4x?x2), 0 <2: < 2
0 otherwise
(1?) What is the value ofc ?
(ii) Find CDF OfX.
((1) Suppose the joint denSity of X and Y IS giVen by
4y(x? ?y)ef(?*7): 0 < x <_ 90 0 < y- < x
f (1' y) .{0 ; otherwise
ComputeB[X|Y '7] . . ._ :.. . 1: ~-
(e) If X and Y are lndependent PoiSSoii randon?1 variables with?
Calculate the conditional expected value of X giveh that X +' "
, (f) Three white and thre biack balls are distributed in two unis. 131161111 ~111/11?5'3?311111?"~ea'9'11'-
' contains three balls.-.?: 59')? that the system is in state i, i =1)"- 1, 2, 3 if. theg?rst 9111
contains 1' white balls; ?At aeh step, We draw one ball ?om each urp _and place the bill drawn
from the ?rst 11m int9 the seeped and conversely with the bail from the second um. Let X_..
denote the state of the 9 111 after the 11?? ste p. Calculate the tmnsiti'o?h prohabihty?matnx
. (g) Data was eoliected. 911 1'1- =iSh'e'ar force (kgS) and y percent ?ber dry weight to test the
? toughhess a11d ?brouSness '9 aragus as a ttiajoi' _cietermman 9f quality. The followmg
observations were recorded 5? ' "
?=18'ZX1=195-0 Zx -25197o Zy,= 47 .92, 2y} =130 6074 Ema 5530. 92
Calculate the va1ue of sample correlation coe?icient and else compute the coef?cient of _?
_ deteiinination. Intexpret the results
1?
Q. 2(a) B111 and George g9 target sh991i11g together. Both Sh target at the Same time. '
. Suppose Bill hits the' target with probabihty 0. 7,- whereas Georg Independently. 11113111: target
_ with probability 0 .4...
"(i) Given {1191 exactly 9119 91191 1,111 the target, what? 15 theprpbab 'tythat it was George s 51191?
(ii) Giyen that the target fut, what 15 the probability that (39ng ? it?
.(b) The dice game craps is played as folloWs. The player thr9ws tw9 dice, 111111 1f1he sum is seven
' or eleven, then she wins. If the sum is twp, three; or twelve, then' She 191195, If the sum is anything
else, then she contim'1es 1111de until she either throw that number again (19- Which case she
wins) or she thmws a- Sev'en (in which case she loses). Calculate the prpbability that the player.
wins. 1
3
Q.3(a) Suppose that two teams are playing a series?of games, each of which is independently
won by team A with probability p and by team B with probability 1?p. The winner of the series
, k is the'fust team to wm four games. Find the expected number of games that are played.
(b) If X and Y are independent gamma random variables with parameters ((1, 2t) and (B, X),
resPectively, compute the'joint density of U = X?+ Y and V = X/(X + Y).
? t2.4(a) De?ne Moment Generating Functiqn (MGF). Obtain MGF of normal distribution and
hence ?nd its mean and variance.
.(b)..Statc and prove Markov?s inequality and hence derive Chebyshev?Is Inequality. ?
? Q.5(a).,A pgisofter is trapped in a cell containing threedoors The ?rst dob: leads to amnnel that ?
'- returns him to his cell after two days (if travel; The second leads to a tunnel, that returns him to
3" his cell a?erthree days of travel; 1113311361 d'oor leadsjgnmediately to freedom; Asgttme? that the
' ', prisoner will always select doors?l,12~?la_hd?3 with pmbability 0,5, 0.3; 0:2, what is. the expected
'f' j ' . transient or remnant; _
V Q?mbCf of days until he reaches?fr'eedom? Fi?d the variance of the numbei' of * days until the
f-prisoner reaches?eedom; ' ; z ' = ;. ' ~ ' i. ? I '
(b) -~If X and Y are independent tandoutvatiables both uniformly distributed on
(0, ?1), then calculate the probability density of X + Y.
Q.6(a) Specify the?c1a35es of the fo?bx?ipgiMarko'v chains, and determine Whether they are
NIHNIF'd-
NIP Q NIH
O NI?-NJH' .?
(b) Fer the following?transition'?rqbabilitxmat?x efa three state maxkovliehain, in the long. run
. i ' ? .5 0.4 701:1} :?j y. ' ?
_~' 0.2 03 .05
. Q.7(a) For any random variables f?-Y. Z arid c'enjstant a,brove that
(i) Coven?) ='-.c- Covom; . . _
(ii) Covoi. Y + Z) _= COVOC. Y) + Cov(X.Z).
v _ . Mom-
?i .
(b) Using the principle of Lcast square, ?t a straight line to the following data:
X 5 12 l4 17 23 30 40 47
.Y 4 10 I3 ' 15 15 25 27 46
This post was last modified on 31 January 2020