Final Exam with Answers - Probability and Statistics in Engineering I | IE 23000, Exams of Probability and Statistics

Download Final Exam with Answers - Probability and Statistics in Engineering I | IE 23000 and more Exams Probability and Statistics in PDF only on Docsity! IE. 230 — Probability & Statistics in Engineering 1 lene Koy Closed book and wares, /20uirmes. L Tros or false, (foreach, 2 pointe if conrest, | point if left blank.) (ab T(t) ‘When choosing a random sample without replacement from a finite fon, it is possible that snene member of the population is chosen twice. “) F For any randocn variables Xp sed XN, the following resule is true: eX ah = BX |) + BX g). teh, T (F) A point estimator is often a single number, but il can be an interval i 7 © A paint estimator is said to be “maximum Likelihood” if it is equal wo the Largest value in the sample. th oT 2 Chebyshev's Inequality guarantess that the sum of many random variables is (ac least approximately) nocmally distrilxrted. cy) F The standard deviation of a point estimator is called its standard error, ct? FP Wf @is an unbiased estimatee, then the mean squared error of @ is equal to the variance of @. ih) T &) Mean squared error is one, bur not the only, measure of 2 confidence inberval’s quality. acter Let Z denote the sample mean of a randam sample af size. ‘The standard espe of ¥ decresses as the sample size a increases, Gy T (©) the ith obeervation from a sample of size m ie called the ih order stacistic, ra] ft) FA statistic is 2 function of the observed sample values. oT ( If (X, ¥) bas a bivarinte normal désiribution, then P(X <0, ¥ < 0}=0. mf) F if Xand Fare independent, then coma, Y= 0. of) ® Let X have @ binomial cistritetion with = 4 and p= .2 Without the inuity conection, the normal appronination to the binomial woukd yield PIX=3)=0. Final Exam, Fall 1999 Page | of 8 Schmeiser od IE 230 — Probability & Statistics in Engineering 1 lg Name Fs ‘2 Let X denote a normally distributed eundom variable with meen phand standard deviation o. ‘The peh quastile of X is the constant sp that satisties PUY Sap) =p. When p= 0 and = 1, the comesponding value is demaied by 3p. be (a) Sketch the density function of X. Label both axes, Seale the horizontal axis. Et : . : + : 7 Sy pe wet Me wer pt a “A {b) Circle the largest value. (Anse depends re the Xa xe ‘a9 volwaw'a $ + are & * fe) Circle the trae a © withen Oo % " Re 0 o=ty-W70) qantas, tp Xp 7 aabs 3. Throughout this course we have followed a consistea notational convention that indicames {,, the nature of vanous quantities. Lerecribe: exch expression below by writing “randoen ks e virial”, “event”, *comsant™, or “undefinex!” on the blank limes. ea carn? _ BY by <X _ vent {c} Vae(¥ < ¥). Hodedi ned (aE _ Comstant fea constant (oo __éenstant tghPix = 3) toms fan Firval Exam. Fall (994 Page 20 & Schmeiser IP 230 — Prosabiliry & Statistics in Exginesring | UE rane probabitity that «bit has high, moderate, of low distortion coon respectively. ‘Suppose that three bits are temsminied ame) bat thee ptt each bitis independent of the other bits. § b (a) Consider # fourth outcome: thal a bit hes mo destoriion, the first bit hss 10 distortion? Fi ne disterBen\ Z f- dai-0.0f- O18 40 a=plas ahitertian\ =p—— cé (b) What is the probability that exactly two bits have high distortion and ane has moderate distortion? 3 # eaat plie?, tort, DMG, Jed CoD C45) = a leit lot) = ,¢oa0!2 +~——_ betwee (KK, a7) paceman Let esate, bet fe) What is the expected number of bits having low distortion’? E(X)* Ps 23015) 2239s C * (d) Comditional that che finst bit has low distortion, what is the peobability that the second I bit has high distortion? 0.9 52—-_~ becaure bite ane nda pensbcKe Final Exam, Fall 1999 Page Saf 8 Schmeiser TE 290 — Probability & Statistics in Engineering I td nae __Key 7. (Moatgomery and Bunger, 3-106.) Sappose that a ket of washers a¢ large enough that it ean be nssumed that the sampling is dese with replacemest, Assume thet 60% of the washers exceed a target dhickness. Let Aj denote the event that washer / exceeds the ‘eesti Given! PCA.) 2G foe cei bee ta} What is the peobobility that a randomly selected washer does not excord is target thickness? PUA > (- PCA) a joj gt ff A (b) Write the event (not ite probability) that washers 1, 2, and 3 all exceed the target oF thickness. AAA AAs et fe) What 6s the mingmum number of washers, «, that need to bss selected so that the me ppeobability hat all the washers are thinner shan the target is teas than 0.107 plaiaa) fA) <o.¢0 plas) Pla)» PAL) © 40 Conbylll => (A <a00 C Patt =) = ath hin -#) =p ns3—— = sf {) Circle the correct answer, The first sentence: of this peoblem (whee "with replacement” is ssaumedi} affects the answer to Pact(s} jew a oo eda ; Ae placemest” Se indapfemce Final Exam, Pall 199% Page bof E Schmaiser da) by . i tea URE EE 1.230 — Probability & Statistics in Engineering | \P> Name __ 3, Let X denote the result of rolling # six-sided die; that is, X is the number of dots facing up. Assame that all six sides are equally likely. c¢é fa) Write the mass function of 7 {Be complete - ie of ete SES Eb $x) i ad otherwise tb) Pind By?) eeepc yt £C)* | i Z . p> = (S¢ t¢ fe) Find the conditional mass function of X given that A <3. (Be oa e 23 = E e242) arlPrnae bade fir ; _ PCkayfed es (Z5(yer)X )* PURER) - plSr+) 7 Fee i : af oe AA Busiapy (evld)--+ C4) CA Cay Pind Bx |X <2, : Niza PCx<=r} Kez)2! = oe | xe =—— Final Bram, Fal! 1999 Page 7 af 8 ‘Schnneiser