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

Download Probability and Statistics in Engineering I - Final Exam with Answers | IE 23000 and more Exams Probability and Statistics in PDF only on Docsity! IE: 230 — Profability & Statistics in Engineering L Ldn Koy Closed book ond mates, | 20 animes. L True or false, (foreach, 2 poate if corest, 1 paint if beft blamk.) ( T, When choosing @ random sample without replacement from a finite fom, it is possible that some member of the population is chosen twice. a F For any random variables X, aed Xi, the following result is true: 4X2) = BiX, ) + Big). teh, T (®) A point estimator is often a single number, but il can be am interval 7 &) A point estimetor is said to be “maximum likelihood” if it is equal wo the largest value in the sample te T ® Chebyshev's Inequality guseuntess that the sum of many random variables is (at least approximalely) noermally distributed. oT) F ‘The standard deviation of a point eslimabor is called. its standard error, aD F Of Gis an unbiased estimator, then the mean squared error of @ is equal to the variance of @. ) T(E) Mean squared eor is ane, but not the only. measure of 2 confidence interval's quality, a F Let denous the sample mean of a random sample of size n. The standart esror of X deoressss a5 the sample size u increases, Gi) T (F ) The Ath observation trom n sample of sine we is.called the éeh onder statistic, (kh (te A statistic is 2 function of the observed sample values. or we, ¥) has a bivariate normal distribution, then P(X <0, ¥ <0} =0 im (0) F if Xand Yece independent, then earx, ¥) =O. oD Let X have & binomial distribution with a= and pe =.2. Without the ity comection, the normal approximation to the binomial would yield PX =H=0. Firsal Exar, Fall 19° Page Lof a Schmeiser it IE 23 — Probability & Statistics in Enginecting 1 \g Nome Keng ¢ 2, Lee X denote a normally distributed eudam. variable with mean po and standard deviation a. ‘The pth quaetile of X ls the constant xp that satisties P(N Sx,) =p. When ji =O and woe 1, the comespanding value i dezented by 2,. o¢ (a) Sketch the density function of X. Label both axes, Seale the horizontal seis, pede aT ff ar poe af dh) Circle the largest value. (Anse depends om the ‘op (0) & rohnert FS = of) bye‘) tee thera aero aither“¢ fin t= bay — Uh onto, % 4p tp ae 3. Throughout this course we have followed a consisteat notational convention that indigmes |, the mature: of wanous quamiities. Cerecrite each expression below by writing “randoen variable", “event”, “constant, or "undefined" on the blank lines. ks ott a fay x? BY (bp PaX _ Event te) Vent =x), _tadeti ned — i) Exe) __ Comstant (0 constant ne fon stan me fa} POE <3} __tms tant Final Exam, Fall [440 Page 2af & Schmeiser IF 290 — Probability & Statistics in Engineering 1 \ Lam 6. (Montgomery and Runger, 6-32.) In the transmisson of digital probohifity that s bit hos high, mederare, o¢ low distortion goon respectively, Suppose that thro Brits are trarsmitted wed that tbe th ech bit is independent of the aber bits. € b (a) Consider & fourth omicome: that & bit has me distortion, Wit is the poobabality Tha fr the: first bit hed no distortion’? Pt ne dz see a 2 {- al-0.0f- 0.75 40 == Pp bao divtert jae | =. cé (b} Wher is the probability that exactly two bits have high distortion and one has moderiie distortion? 2 plX= Zz, ¥, a I, x =9) = iS A é oi) (o#) é. 45) 3 leit Cor) _soaoll +—__ because (KX. 3) fa wie [Ogata 4 ge) Whar is the expected number of bits having Low distoution'? bee E(X)* Ps 2301) 2 2.8#5———_— t 4 (4) Comstional thatthe fe bit hs low distatien, what isch probability that dhe second f ‘bit has high distortion? 0.9 Ss hecawre Life eee inde pense TE Final Exam. Fall 1999 Pape Soff Schmeiser TE 230 — Probability e Statistics in Engineering I (d vom Key T. (Moangerery and Runger, 3-106.) Suppose that bot of washers ia large enough that it can be nesumed that the sampling is dose wiih replacemesit, Assume thet 60% af the washers exceed a target hicknett. Let Aj denote the event that washer (exceeds the target thickness, ‘ . “ts 7 = Given: PCA.) iG tee tei tee & hs (a) What is the probability thar a randomly selected washer does not excocd ia target - thickness? PUA) = (- PCA) é (by Write the event (not its probability) hat washers 1, 2, und 3 all expec the target oe theckness. A,A Ay fax Ve (oc) What és the minimum number of washers, «4, that need to be selected so. that the peobability that all the washers are thinner than the tarpet iss bes than 0.007 PAAR VA) £0.00 septa) PA PAd) £0.00 ara) =e 6A <al0 Cpt) = A ath win -#) —_> ne 3 5% 0) Circle the comect answer The first sentence of this problem (whose “with i replacement! is ssaumedt) effects the answer i Part(s) a & (Om Ge) we (be) ¥ RM fa facie meat ‘= Pndlapewflence Finall Exe, Fall 1999 Prage 6 nf & Schmciser .. rie tecceeanay WOE & [F220 — Probobility & Statistics in Engineering! ‘Pr Name X24 = Let X denote the result of rolling # six-sided die: that is, X is the number of does facing up. Assame that all six sides are equally Wkely. c fe (a) Write the masa function of N? (Be complete.) fl) = ie pF erheStS & 0 epee 4 thy Find Bue?) Ea) (A .2ld)s-+ cE) tf (¢) Find the conditional mass function of X given that X <3. (Ba oa e oe Kell? panne bab [ZX Pky ES welX O° “Saray = P(%r) _ “Ce 4D | ee ae Ley geht ae = ct (a) Pind Bie |X < 2) Yer => plx=s} Lezel => £(x|x<2)=!1+-— Final Exam, Fal) 1999 Page 7 af ‘Schoneiser