midterm solutions - Ah/g KM 1. Consider a probability...

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Unformatted text preview: Ah/g KM 1. Consider a probability density function shown below. (Assume that the probability of any value outside the range [0, 2] is zero.) a. What is the formula for this PDF? b. What is the expected value of this PDF? c. What is the second moment around zero for this PDF? d. What is the variance of this PDF? e. Given a sample of 64 independent observations drawn from this distribution, y.,y2,...,y64, what is the approximate probability that the sample mean, 7, is less than 1? e3 HM < szsX-‘E—y <l/37\ ‘ 0/3? 2. The net change, C, in riders at a particular bus stop is the difference between the number that board, B, and the number that alight, A. Suppose the latter two variables are both Poisson distributed with parameters [1,; and AB, and independent. a. Let fc (x) be the pdf for C. Write an expression for fC (0) as a function of AA and 13. b. Write an expression for E (C) as a function of AA and 13. c. Write an expression for VAR(C) as a function of AA and 13. d. Suppose we have some observed values for C, c], ...,c,,, but not for A and B. What are the method-of—moments estimators for AA and AB? Hint: for full credit, neither of the expressions for b and c should contain a summation sign! C: ETA B~Pon50e7 A“%&(Afl, AL% 00 C=O=€>~A A=B whm C=0. J: ZL ’AA AZIeHAB 62(0): % AA o BZL‘ b7 Ed: Etta-A) : Egg/EA; AVA], XNPmsKM BOW/txrfi fl Vol/NC? = van (3A) 1 \lm(€>*(’AW r” VOAl5+ VON-A) : WBMvDWa/KA} " vaAvaaAA 3 agar Ivan N):van()(.+XD : vomXHvo/iXL id) 1% Mai/1W figx-fi Em: E: Alb-N 2nd \\ 7L : E(C‘L): C} =’ \{0\/\C4(EC)L l ELI: AA'lWB’WEY— Q tom‘s , 2 MflkleAM/I . E=Ayai * F {at : "ANN 6+ CT—(ElZ = 3N5 av amaha, AA: A's—E : étiltEt—E) 3. Consider two linear cities of lengths L [=2 km and L2=4 km arranged as shown below: C1ty‘l C1ty2 : a. The distance between the eastern (right) end of City 1 and the western (left) end of City 2 is 1 km. Trips from City 1 to City 2 are equally likely to originate anywhere in city 1 and terminate anywhere in City 2 and the locations of the two trip ends are independent. b. Consider an experiment where, for a trip from City 1 to City 2 chosen at random, we observe the trip start point and end point. Depict the sample space in two—dimensional space where one axis corresponds to the start location in City 1 and the other to the end location in City 2. c. Identify the set of points in the sample space where i. The trip length is 1 km. ii. The trip length is 7 km. iii. The trip length is 3 km. iv. The trip length is 5 km, d. Find the PDF for the trip length.. e. Plot the PDF, and show how one would calculate the CDF using the plot. c) '0 704M WW: I km 10/)(57 KM . M tram 0m 2 ’ (cm/«Jr loin) } (7v 0) (177’ /’ / —+ c. 1 (gm l1") L, W lw 0) 011’“ am l N) ’X= 5 km (bhld 19w léx<$ (0’0 0:0“? 62’" (5;: i=5oi‘vé 06% I“? :7 {a} “191%; 5gsz (5,74) i=5x+ib (7,07 0:7(wés F):—WOL :7 #:5XJIOL:_QO( oz: vé fifivhzfl'lz : :3 W70: 7/8 ‘éx My» *zévc-J léVSB a: 3<7¢S< jg'JgX 5<V§7 {(x) ‘ 6) PDF {9101 (see pm m pm d) FM: f‘WrJ dwfjdw 5,155“?de l 5 g 5 q 5 //Umngw)k* PAM Pam 1, r x (Y)- 1 3:14;“ 8 tags? X 73 _ : L317??de 341% d 3&5 v '. 'u x :DYY 'xdxifgg’dd“ ‘L jg-g—éxow Lj 5<X<3 4. A rural arterial has a series of n signalized intersections far enough apart so that whether a particular motorist must stop at any one of them is independent of her need to stop at any other. Moreover, this probability of having to stop, p, is the same for each intersection in the series. A sample of two motorists is observed, and the number of intersections at which they have to stop is recorded as l and 2. i. What is the MLE for p, assuming that the series includes 3 intersections? ii. Is this estimate biased or unbiased? iii. Suppose we want to use the data to estimate both p and n using MLE. What are the likelihood and log-likelihood functions that we would use? iv. Of the values 3 and 6, which is the better estimate for n based on the likelihood criterion? (Hint: For a given value of n, first calculate the MLE for p, and use that along with the n value to evaluate the likelihood.) D MLE Fm p fi—hmes wmimsstop w PJMUWQO m VL:3, 61W LLP): (il P“1>72*(§l F20??? Cl yup? bwwr lax/Low MLEQN P to A:%; gift: :>$;__L m P v w w m Pm-M w-w mm m w : (n1)L 3, 2n}. (th'a‘ln—Dl P LUMP? =llfl (0‘3— ln [MN] - \v‘ [2:]- lnf(n—7_3‘.'l + 3MP + OWE-D Ina-p) N3. L(?>,Pl given m t!) 9 IPZi. mm = QC—W’m‘ m1.” : 0M [Mm n=(p, ii: A“ Z: LN» '44): t A (—3 ‘3. 9W3} ) (Il(4l 4) ( (Li 4) [13,413 L((o,‘/4l) : gonswm“ : 0.10% a) W > 0‘00” :7 Ema meimm-fiw n om “HAW mum/gnu? “flucakQ/L . 5. Peter, Samir, and Milton share a carpool to commute to work. If there is an incident in the carpool lane during the morning commute period, they are late 3A of the time, while if there is no incident, they are late ‘A of the time. The probability of an incident occurring in the carpool lane during the morning commute period is 1/12. If, on a given day, they are late for work, what is the probability that there was an incident in the carpool lane? LC Oom, A: OCUch’V' 9(L1A)?(MHM PCLMWLM * W L l AC) PW) ll ,Jmé. MA... 1:8 + —:.. 0.2M ...
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midterm solutions - Ah/g KM 1. Consider a probability...

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