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STA355H1S Assignment

创建日期：2024-07-20 17:06:51

所属分类：统计学statistics

标签： STA355H1S

Assignment

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Assignment STA355H1S

Instructions: Solutions to problems 1 and 2 are to be submitted on Quercus (PDF files only)

Instructions: Solutions to problems 1 and 2 are to be submitted on Quercus (PDF files only) – the deadline is 11:59pm on April 5. You are strongly encouraged to do problems 3 and 4 but these are not to be submitted for grading.

Problems to hand in:

1.Inclass, we discussed a general class of estimators of g(x) in the non-parametric regression model

of the form

Yi = g(xi) + εi for i = 1, · · · , n.

where we required that w1(x) + · · · + wn(x) = 1 for each x. However, it is often desirable for the weights {wi(x)} to satisfy other constraints. For example, if Var(εi) is very small or even 0 and g(x) is a smooth function, then we would hope that g(xi) ≈ g(xi) for i = 1, · · · , n.

(a)Suppose that Yi= g(xi) = β0 + Σp^βkφk(xi) for i = 1, · · · , n where β0, β1, · · · , βp are some nonzero constants and φk(xi) are some functions. Define the n × n smoothing matrix

Then

If g(xi) = g(xi) for i = 1, · · · , n for all β0, β1, · · · , βp, show that

are eigenvectors of A with eigenvalues all equal to 1.

(b)A smoothing matrix A typically has a fixed number r of eigenvalues λ1, · · , λrequalto 1, with the remaining eigenvalues λr+1, · · · , λn lying in the interval (−1, 1). Usually, by varying a smoothing parameter, we can make λr+1, · · · , λn closer to 0 (which will make g(x) smoother) or closer to 1 (which can result in g(x) being very non-smooth).

If A is symmetric then A = ΓΛΓT where Λ is a diagonal matrix whose elements are the eigenvalues of A and Γ is an orthogonal matrix with Γ−1 = ΓT ; we will assume that Γ doesnot depend on the smoothing parameter (i.e. the eigenvalues of A depend on the smoothingparameter but its eigenvectors do not). In lecture, we gave the bias-variance decomposition

The decomposition A = ΓΛΓT make the effect of the smoothing parameter on the bias and variance components very transparent.

Show that as λr+1, · · · , λn shrink towards 0,

(i)“Aε“2= εT AT Aε decreases;

(ii)“(A−I)g“2increases if λr+1, · · , λn ≥ 0 unless g is an eigenvector of A with eigenvalue 1.

(c)In practice, the form of the smoothing matrix is typically hidden from the user and in fact, is often never explicitly computed. However, for a given smoothing procedure, we can recoverthe j-the column of the smoothing matrix by applying the smoothing procedure to pseudo-data (x1, y1∗), · · · , (xn, yn∗ ) where yj∗ = 1 and yi∗ = 0 for i j.

Consider estimating g using smoothing splines in the model

Yi = g(xi) + εi for i = 1, · · · , 20

where xi = i/21. The following R code computes the smoothing matrix A for a given value of its degrees of freedom (i.e. its trace) and then computes its eigenvalues and eigenvectors:

> x <- c(1:20)/21

> A <- NULL

> for (i in 1:20) {

+ y <- c(rep(0,i-1),1,rep(0,20-i))

+ r <- smooth.spline(x,y,df=4)

+ A <- cbind(A,r$y)

+ }

> r <- eigen(A,symmetric=T)

> round(r$values,3)

[1] 1.000 1.000 0.913 0.579 0.262 0.115 0.055 0.029 0.016 0.010 0.006 0.004[13] 0.003 0.002 0.002 0.001 0.001 0.001 0.001 0.001

The matrix A is by definition symmetric and using the option symmetric=T in the R function eigen assures that the eigenvalues will be computed as real numbers (i.e. no imaginary component). (Note that A has 2 eigenvectors with eigenvalue equal to 1; this means that the smoothing method will recover linear functions exactly.) Repeat the procedure above using different values of df between 3 and 10. What do you notice about the eigenvalues as df varies?

(d)The smoothing matrices in part (c) are not only symmetric but centro-symmetric; that is, if

Suppose that v = (v1 v2 · · · vn)T is an eigenvector of A. Show that v is either symmetric (i.e. vj = vn−j+1 for all j) or skew-symmetric (i.e. vj = −vn−j+1 for all j).

(Hint: Define J to be the matrix that reverses the elements of a vector:

A is centro-symmetric if AJx = JAx. It suffices to show that if v is an eigenvector of A

then Jv is also an eigenvector.)

(e)(Optional but recommended) Take a look at the eigenvectors of A in part (c) (for a particular value of df). This can be done using the following Rcode:

> r <- eigen(A,symmetric=T)

> for (i in 1:20) {

+ devAskNewPage(ask = T)

+ plot(x,r$vectors[,i],type=”b”)

+ }

(The command devAskNewPage(ask = T) prompts you to move to the next plot.) Note that the first few eigenvectors are quite smooth but become less smooth.

2.Supposethat D1, · · , Dn are random directions – we can think of these random variables as coming from a distribution on the unit

circle {(x, y) : x2 + y2 = 1} and represent each observation as an angle so that D1, · · · , Dn come from a distribution on [0, 2π).

A simple family of distributions for these circular data is the von Mises distribution whose density on [0, 2π) is

where 0 ≤ µ < 2π, κ ≥ 0, and I0(κ) is a 0-th order modified Bessel function of the first kind. In this problem, we want to derive tests of the null hypothesis H0 : κ = 0 versus the alternative H1 : κ > 0. Note that under H0, D1, · · · , Dn have a uniform distribution on the interval [0, 2π).

(a)Suppose that we want to test H0 : κ = 0 versus H1 : κ > 0. Consider a likelihood ratio test of H0 : κ = 0 versus H0 1: κ = κ1 > 0. Show that this LR test rejects H0 : κ = 0 for large values of the statistic

where µ is the MLE of µ (derived in Assignment #3).

(b)TheRayleigh test of H0 : κ = 0 versus H1 : κ > 0 uses a test statistic closely related to the LR statistic in part (c). Define

Show that the limiting distribution of R when H0 is true is Exponential with mean 1. (Hint: Use the bivariate CLT to find the joint limiting distribution of the random variables inside the parentheses.)(c)Atest of H0 versus H1 should be invariant under shifts of the angles D1, · · , Dn; in other words, if T = T (D1, · · · , Dn) is a test statistic for testing H0 then for any φ, T (D1, · · · , Dn) and T (D1 + φ, · · · , Dn + φ) should have the same distribution under H0. Show that this invariance condition holds for the tests in parts (a) and (b). (Hint: It’s sufficient to show that T (D1, · · · , Dn) = T (D1 + φ, · · · , Dn + φ) for any φ.)

(d)The file txt contains dance directions of 279 honey bees viewing a zenithpatch of artificially polarized light. (The data are given in degrees; you should convert them to radians.) Use the Rayleigh test to assess whether it is plausible that the directions are uniformly distributed.

Supplemental problems (not to hand in):

3.Supposethat (X1, · · , Xn) have a joint density f (x1, · · · , xn) where f is either f0 or f1 (where both f0 and f1 have no unknown parameters).

We put a prior distribution on the possible densities {f0, f1}: π(f0) = π0 > 0 and π(f1) = π1 > 0 where π0 + π1 = 1. (This is a Bayesian formulation of the Neyman-Pearson hypothesis testing setup.)

(a)Show that the posterior distribution of {f0, f1}is

π(fk|x1, · · · , xn) = τ (x1, · · · , xn)πkfk(x1, · · · , xn) for k = 0, 1

and give the value of the normalizing constant τ (x1, · · · , xn). (Note that π(f0|x1, · · · , xn) +π(f1|x1, · · · , xn) must equal 1.)statistic代写

(b)When will π(f0|x1, · · , xn) > π(f1|x1, · · · , xn)? What effect do the prior probabilitiesπ0and π1 have?

(c)Suppose now that X1, · · , Xnare independent random variables with common density gwhere g is either g0 or g1 so that

fk(x1, · · · , xn) = gk(x1)gk(x2) × · · · × gk(xn) for k = 0, 1.If g0 is the true density of X1, · · · , Xn and π0 > 0, show that

π(f0|x1, · · · , xn) −→ 1 as n → ∞.(Hint: Look at n−1 ln(π(f0|x1, · · · , xn)/π(f1|x1, · · · , xn)) and use the WLLN.)

4.Suppose that X1, · · , Xnare independent Exponential random variables with parameter

λ. Let X(1) < · · · < X(n) be the order statistics and define the normalized spacings

D1 = nX(1)

and Dk = (n − k + 1)(X(k) − X(k−1)) (k = 2, · · · , n).

As stated in class, D1, · · · , Dn are also independent exponential random variables with pa- rameter λ.

(a)LetX¯n be the sample mean of X1, · · , Xn and define for integers r ≥ 2

Use the Delta Method to show that √n(T − r!) −→d

N (0, σ2(r)) where

σ2(r) = (2r)! − (r2 + 1)(r!)2

and so √n(ln(Tn) − ln(r!)) −→ N (0, σ2(r)/(r!)2).

(Hint: Note that D1 + · · · + Dn = nX¯n. You will need to find the joint limiting distribution of

and then apply the Delta Method; a note on the Delta Method in two (or higher) dimensions can be found on Quercus. You can compute the elements of the limiting variance-covariance matrix using the fact that E(Dk) = k!/λk for k = 1, 2, 3, · · · which will allow you to computeVar(Dr) and Cov(Dr, Di).)

Note:

We can use the statistic Tn (or ln(Tn) for which the normal approximation is slightly better) defined in part (a) to test for exponentiality, that is, the null hypothesis that X1, · · · , Xn come from an exponential distribution; for an α-level test, we reject the null hypothesis for Tn > cα where we can approximate cα using a normal approximation to the distribution of Tn or ln(Tn). The success of this test depends on Tn −→ a(F ) where a(F ) > r! for non-exponential distributions F . Assume that the F concentrates all of its probability statistic代写

mass on the positive real line and has a density f with f (x) > 0 for all x > 0; without loss of generality, assume that the mean of F is 1. If k/n ≈ t then Dk = (n − k + 1)(X(k) − X(k−1)) is approximately exponentially distributed with mean (1 − t)/f (F −1(t)), which is constant for 0 < t < 1 if (and only if) F is an exponential distribution. Since the mean of F is 1,

By H¨older’s inequality, it follows that

since we assume that the mean of F is 1. Thus a(F ) ≥ r!. Moreover, if a(F ) = r! then (1 − t)/f (F −1(t)) = 1 for 0 < t < 1 or (1 − F (x))/f (x) = 1 for all x > 0, which implies that F (x) = 1 − exp(−x).

(b)Usethe test suggested in part (a) on the air conditioning data (from Assignment #1) taking r = 2 and r = 3 (using the normal approximation to ln(Tn)) to assess whether an exponential model is reasonable for these data.

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IAB320 CHC5028 TSP ICC565 CS512 320SC COMP2396 COMP207 SVM BSCH EECS 2101 COMP3230 CEG3136 INFO 3440 EE3009 MGT7174 COMP 3334 ECON20120 Econ 702 CFRM 505 CS4001 CSE 328 AMATH 482 ENG2008 MATH39522 DTS101TC CSC318 EDUC 124 NBA 5110 ECON0125 ELEC7310 EEP101 ECE 777AE MDS 6112 Bioc0036 ACCFIN5246 STA 404 CFRM505 Chinese 2241G MATH 376 AUENV 218 ELEC0006 ACCT 3230 FIN3020 ENVM3115 CMPSC 464 EE 4233 CST 8284 ENGI 2211 EEB240 COMP52715 PHYS1120 ECON0019 MAT187 FINN1037 CSci 5421 7SSMM803 STAT 400 LUBS5996M ELC5216 STATS 4A03 ENGG1003 BADM 3661 DATA423 ST332 ECO2121 6406ELE IB3K20 CSE 465 CSE115 CMPUT 206 ecs150 INU1111 QM222 DTS204TC MATH2319 CPT108 SENG6110 MATH3041 INFS3604 MA 4704 CIVIL 750 EEEE4123 CS 3800 comp3511 MEC206 CSCI-UA.202 IOM 302 NBS8295 BMS 201 ECON 3018 ESE 605 ADM3225 STAT7038 Economics 409 ECON 503 MSE800 4SSMN902 IEOR E4106 FRE6831 ISE 563 MATH 451 ECON0001 COMPSCI369 CITS4402 MA1608 MQBS1030 MUSC1982 CITS4012 ACC202 BAFI1026 MA117 ECON5074 MN-3507 ECON-UA 11 AC3059 FIT3094 STATISTICS 260 EC333 MUSIC 2F03 BPLN0025 Math 632 ACCT40115 EC3044 EGEC 540 MATH 521 MATH 131B Math 164 PHILO225 ACST2001 CSI2108 ECE 5041 MSINM014 ECON60822 MATH350 EPPD1063 ECE 3561 STATS 101 MATH2014 ECON207 WEBSITE EEC180 ACCT2343 COMP24112 MATH21112 STAT141B ECON7230 CCT109 H5F COSC2531 MGT 180 ECO 2173 CSE 30 ELEC9714 DPBS1120 SMM069 MVP CV4 7AL SDOF SYSTEMS Architecture SQL DS1000B STAT 3701 STA 442 GU4206 ECON 331 CSCI 3120 STAT600 ANLY-511 STAT2402 CSE 2011 GR4206 Stats 20 SCIE4401 STAT464 CSE 230 Network Econ 3818 SE-555 Applied CSCI3180 FIT5197 Robotics 41013 CO004 CS 200 MANF9472 CITS1001 CSE 320 COMPSCI 335 CSCI 3022 OPIM 3220 CSE278 ADM3308 STAT 467 STATS 731 CMSC 216 Mathematical PHYS 454 MAST 333 CSE20W21 MATH2302 MF 796 CS 2100 Mathematics Math 3FF3 CVEN2002 MATH-UA.0252 COMP344 CS631T KNE488 ETF2700 CS314 EGB242 ECE 2671 AMAT 592 MAT 581 COMP1037 ECO £404 ECE-5200 Maths CSC570 COMP284 CSCI3287 EER CS BDT5002 FIT9130 COP4600 CS348 CHI2550 BIA-656 Comp 271 CSCD43 SQL code INF 559 COMP5434 MAE384 Stat 384 Microeconomics Econ G25T Econ 331 STAT 252 ACFI3422 STA 104 ECON 4261 Statistics 305a ECMT 674 Markets Statistics 506 Statistics 4205 ECO6133 ECON 327 Stat 525 Econ 107 Trade 178 Financ CGE13111 ECON 308 MANM301 FIN 450 INVESTMENTS STA355H1S Economics 454 Networking STAT375 IC205 ECON 4850 MSCI 523 NN5-096 BMAN 20821 STAT0029 FM50 ACCT 374 Digital stock MgtF 405 COMP0040 STA414 BUS10306 Comparison CGE25111 EF 5070 GGR321 EC471 EF4321 Econ 101 ECO 404 ANLY550 CMP2092M CSC8111 CSCI 520 Shooter COP5612 FINA 6421 CSC8001 CSCE 101 Software EECS 12 CS2105 COMP 204 EECE 5642 Final ISTA 131 STAT 7008 CSE 468 MATH1058 CSCI3104 COSC1073 CS 112 CSC 1011 CSE105 JSP CP1802 ACS CS2230 CO2017 EECS 233 CSC 656 INFS1609 CPS 350 EBU5502 AACS2204 CO7506 TCSS305 CPS 209 CSC 220 CSE 216 CM3114 CSE 17 CS2C CSE3320 ICT162 COMP 348 EECE 2160 CS 381 CS 3305A CIS 657 CMP2090M CSCI251 IY3840 CSCI-UA.0480-11 COMP604 DSA 5005 COSC 3319 CSE 270M CIS 3207 Programs CSCI 2132 CS270 CSE-381 COMP3230A CIS 212 CSI 333 FINM 326 FINM-36702 Introduction to Software Analysis EEE102 GNG1106 ECS 60 CS 161 CSCI851 CSE1222 EDPS0249 TCHR5003 ANTA02 BBUS1SBY SOSC 1140 MDIA2091 FINS3641 INSY 661 ITEC3040 CPSC 481 computer DEE3519 COMP304 CSE536 Microtheory OFFICE 280 CIVE50003 COSC2673 ECS784U CMPSC/MATH 455 AMA 505 CSE 158 BUS1040 FIT3179 FIT5202 Math 104A Math 111A MBUS107 COMPSCI 761 COMP3710 COMPSCI 316 COMP559 COMP557 CP1 1902 COMP 310 ECED 3403 COMPX523 AGCM 3103 EMET 7001 ENGR3821 CISC 124 IPC comp20005 CS 323 CS105 STAT 326 MATH4007 CS4551 COMP1000 ACIT 4640 ENVX3002 CS2034 FC712 algorithms CSCI 4155 CSI4142 POL 850 MATH20811 2B03 CSCI803 Economies COMP 2016 Architect TRADE INFO1110 FIT1043 networks Robotics ICS 31 ECEC-301 FIT2099 CS 3640 CSE1010 Computational COMP1001 CSE 231 CS10 FIT5166 CSE 400 CSCI561 Math 350 CO-496 ENGG6400 CSCI 1133 COMP3055 MATH 819 CSE 6363 CIS 555 CS4420 COMP0037 ECS 140A Spanning Tree Protocol network Statistics 100B CMPSCI 383 CS 659 CSC 120 COMP 74 CSCV 460 CSCI 2121 ENG3047 CSE 491 PHAS0020 MATH 185 MSCI 517 CS201 CSE 220 COMP 527 CS602 MTH 496 CS909 CS 116 AMS 315 H64RFL/P MGTS3603 ARTS3463 FIT1045 PL241 EE5434 Coding MSF 503 COMP255 SEC204 GNBF5010 JIT104 ECE241 CS 545 COM4509 STA 3187W ECO-5006A DS 310 CSE205 IB9Y8 CSE 3100 4COSC001W IN1900 MGT 201 CSCI 1100 MAT2040 CMPT310 CIT 592 COMPSCI 753 CS 231 CSc 110 MET CS-677 COMP0027 COMP 5416 CPSC5610 COMP7104 SIT215 CSCI 4146 EE6435 CSc361 MSBA7003 CISC 360 COMP3020J CS 320 COMP101 CIS667 CAN201 6CCE3CHD COMP3008J C200 DAT 500S EMAT10007 ST309 CSci 1100 DSC 20 CSCE 474 INFS2044 COMPSCI 711 PROG2111 COMP2211 INFO6002 COMP524 ELEC 9741 IN3043 CSCI 1300 CS101 CSI2110 ITP4514 ISIT312 MATH1090 MATH2019 DS-UA 202 CS 166 CS265 ECON-100-001 CSOR 4231 ECS759P COMP24011 CS 8803 CS 453 INFO90001 ARTS1240 ECN340 T2187 ECE 522 COMPSCI 367 S6140 MA9070 CSCI567 STAT535 MATH 1021 CS 33400 6006CEM MIS501 CSE514 MATH35062 ECON6101 CP1401 PSY4219 COMP304-23A FIT5222 DSCI-553 SYS5020 INFO 5304 ECON0024 OMP9727 CSC8636 MATH60026 6CCS3ML1 CS5012 AIC2100 CPSC 217 G1109 2XC3 CSE 470 COMP2051 EG25H4 CS 7280 CSC1002 Stochastic CITS5508 CS 2820 IERG 4080 SEHS4515 CSE8313 CPS 400 CSE 3521 CIS 313 IS2150 COMP41620 LP002 KXO205 COM6471 COMP 2140 CS251 ITE3101 CST205 CS 340 CISC124 ECON 128 sCSCI3310 COMP0004 MLE 3 IT114107 CSE-3342 LTC2946 COMP122 CSIS 1119B CS560 CS210 COMP2050 CSCI 340 STT481 Econometrics ECE 6100 PSTAT 170 CS5003 COMP 2406 CSC384 BBU4202 Stat 134 ECON41415 CSE1OOF CP2406 ICT501 ICT CSE 104 CSCI 2300 TAS 100 CSCI 3110 CMPE 223 CSC220 CPT105 CSCI 1110 CM3112 CSC8406 SE 3316A COMP1212 CVEN90063 ECS 32B COM1005 COMP124 FV2204 PROG2005 MA569 INFO151 COMP90038 CS4022 IS71083A COMP 424 COMP4033 EECS 1720 CS 2113 COMP1111 COMP212 INFR11199 COMP5911M ECON 5004 ECMM410 CSC B07 COMP 3430 CSB352 6CCE3ROB STAT5610 PUBPOL 522 COMP2610 FIT5227 BISY90009 COMP0083 NV 684 BUSML 3150 ECN21004 STAT6016 ELEC3705 MATH0033 CS5820 SCC311 CSci 4061 ECON 8820 INT104 ITE3713 CS551J ECM1410 CSMBD21 COMP 2012 CISC221 Code SCC312 COMP639 CMPSC 221 system COMP3217 COMP1721 ENG5056 Math 466 COMP0039 MFIN 5400F FINN40915 EENG31400 MODL5007M EEET 3050 EEET 3050 Quantitative Methods mathematics COMP1921 FIN3309 VE311 ESCI 1908 FIT5057 COSC2446 COMPUTING Math 110A EE 567 MATH10098 CISC642 EE10134 ECE 321 MTH2051 SIT 281 EEE3314 ELEC273 COMP5930M CE322 ELEC270 EE5101 MCEN90008 MTSC 887 Probability ENGN4528 MAE 424 MA2552 EBU5303 CHEM10600 EBU6018 ELEC2103 ELEE 5200 CSE 535 COMP6528 IS6153 IS3S664 BUSANA 7003 QTS0109 ECO3152B ENGR 227 LCSCI4208 MATH32051 ECS776P MGMT3004 N1551 JAVA ROB-6213 TNS3113 BEEM062 COMP47670 4SSMN903 BST 833 COEN 146 BASC0003 CAS EC501 Networks CSIT940 ELEC372 ACCFIN5034 Math 3527 DATA4000 INT102 FND109 MMM071 ROB6213 ETF5650 ECON 445 QMSS 301 Econ 320 PHY 4390 FIN 111 FINN41615 ECS 116 MTH6158 IOE 552 STA 141B Math 413 ECON3016 Physics 4303 Numerical BFF5270 EECS 111 FINANCE 251 COMP814 MGSC 410 ECON0051 EBU5376 STAT6118 ECON 20110 COMU7000 average acceleration BUSI4412 BFF3121 BIT233 ECMM462 ETC3430 DESN2348 INTE2584 ENME502 MA3XJ INTE2627 MA2815 MA3AM L0101 Math 3MB3 CPCCBC4004 EEEM061 0502E220 APPLIED 6SSMN310 ENV 3310 PM2PCOL3 CONS6201 MATH502 FIT5003 EEEN313 MATH3001 MECH3460 ECON30009 COSC1252 Assembler COMP 3023 ENGG7601 COMP20005 COMP SCI 1103 ICSI 402 INF2C CS320 COMPSCI7064 ECE 3220 PROGRAMMAMING MIPS Computer Science 2413 AM3611 MSOA TE2101 FFT CMPSC-121 CSE 109 COMP SCI 7064 HCI CMP-5013A CCN2042 COMP3400 EEEN30052 CO4215 EE101 KIT107 CSE 421 CSE521 MCD4720 EECS 280 MIT 403 GRPC VG101 APiE EECE 1080C FIT3143 CSE 325 ISOM 3029 CS3307 State SIT255 Engineering MQF Modelling ACX3150 SIA322 ACCT90004 COMU7201 MGMT90018 WRIT1000 COMP3702 BISM1201 COMP7505 IDEA9106 CSE3SMT BFC5926 BFC5280 BFC5935 QBUS3330 ECON3700 IFN521 ECON2121 COSC2626 MA413 DDES9903 ENGG1300 COMP9337 CO2201 ACCT2002 ICM289 MMM054 MSBA-204 DIGM707 CSC2250 Econ312 EVAT python CAN404 LUBS2840 CSCI-GA.2250 COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213

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