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COMP9020 Random Variables

Creation date：2024-05-25 16:24:13

Belonging category：计算机computer science

Tag COMP9020

Random Variables

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COMP9020 23T1

Week 10
Expected Values, Course Review
Textbook (R & W) - Ch. 9, Sec. 9.2-9.4
Problem set week 10
1
Random Variables
Definition
An (integer) random variable is a function from Ω to Z.
In other words, it associates a number value with every outcome.
Random variables are often denoted by X ,Y ,Z , . . .
2
Example
Random variable Xs
def
= sum of rolling two dice
Ω = {(1, 1), (1, 2), . . . , (6, 6)}
Xs((1, 1)) = 2 Xs((1, 2)) = 3 = Xs((2, 1)) . . .
Example
9.3.3 Buy one lottery ticket for $1. The only prize is $1M.
Ω = {win, lose} XL(win) = $999, 999 XL(lose) = −$1
3
Expectation
Definition
The expected value (often called “expectation” or “average”) of
a random variable X is
E (X ) =
∑
k∈Z
P(X = k) · k
4
Example
The expected sum when rolling two dice is
E (Xs) =
1
36
· 2 + 2
36
· 3 + . . .+ 6
36
· 7 + . . .+ 1
36
· 12 = 7
Example
9.3.3 Buy one lottery ticket for $1. The only prize is $1M. Each
ticket has probability 6 · 10−7 of winning.
E (XL) = 6 · 10−7 · $999, 999 + (1− 6 · 10−7) · −$1 = −$0.4
NB
Expectation is a truly universal concept; it is the basis of all
decision making, of estimating gains and losses, in all actions
under risk. Historically, a rudimentary concept of expected value
arose long before the notion of probability.
5
Theorem (linearity of expected value)
E (X + Y ) = E (X ) + E (Y )
E (c · X ) = c · E (X )
Example
The expected sum when rolling two dice can be computed as
E (Xs) = E (X1) + E (X2) = 3.5 + 3.5 = 7
since E (Xi ) =
1
6 · 1 + 16 · 2 + . . .+ 16 · 6, for each die Xi
6
Example
E (Sn), where Sn
def
= |no. of heads in n tosses|
‘hard way’
E (Sn) =
∑n
k=0 P(Sn = k) · k =
∑n
k=0
1
2n
(n
k
) · k
since there are
(n
k
)
sequences of n tosses with k heads,
and each sequence has the probability 12n
= 12n
∑n
k=1
n
k
(n−1
k−1
)
k = n2n
∑n−1
k=0
(n−1
k
)
= n2n · 2n−1 = n2
using the ‘binomial identity’
∑n
k=0
(n
k
)
= 2n
‘easy way’
E (Sn) = E (S
1
1 + . . .+ S
n
1 ) =
∑
i=1...n E (S
i
1) = nE (S1) = n · 12
Note: Sn
def
= |heads in n tosses| while each S i1 def= |heads in 1 toss|
7
Example
E (Sn), where Sn
def
= |no. of heads in n tosses|
‘hard way’
E (Sn) =
∑n
k=0 P(Sn = k) · k =
∑n
k=0
1
2n
(n
k
) · k
since there are
(n
k
)
sequences of n tosses with k heads,
and each sequence has the probability 12n
= 12n
∑n
k=1
n
k
(n−1
k−1
)
k = n2n
∑n−1
k=0
(n−1
k
)
= n2n · 2n−1 = n2
using the ‘binomial identity’
∑n
k=0
(n
k
)
= 2n
‘easy way’
E (Sn) = E (S
1
1 + . . .+ S
n
1 ) =
∑
i=1...n E (S
i
1) = nE (S1) = n · 12
Note: Sn
def
= |heads in n tosses| while each S i1 def= |heads in 1 toss|
8
Example
E (Sn), where Sn
def
= |no. of heads in n tosses|
‘hard way’
E (Sn) =
∑n
k=0 P(Sn = k) · k =
∑n
k=0
1
2n
(n
k
) · k
since there are
(n
k
)
sequences of n tosses with k heads,
and each sequence has the probability 12n
= 12n
∑n
k=1
n
k
(n−1
k−1
)
k = n2n
∑n−1
k=0
(n−1
k
)
= n2n · 2n−1 = n2
using the ‘binomial identity’
∑n
k=0
(n
k
)
= 2n
‘easy way’
E (Sn) = E (S
1
1 + . . .+ S
n
1 ) =
∑
i=1...n E (S
i
1) = nE (S1) = n · 12
Note: Sn
def
= |heads in n tosses| while each S i1 def= |heads in 1 toss|
9
NB
If X1,X2, . . . ,Xn are independent, identically distributed random
variables, then E (X1 + X2 + . . .+ Xn) happens to be the same as
E (nX1), but these are very different random variables.
10
Example
You face a quiz consisting of six true/false questions, and your
plan is to guess the answer to each question (randomly, with
probability 0.5 of being right). There are no negative marks, and
answering four or more questions correctly suffices to pass.
What is the probability of passing and what is the expected score?
To pass you would need four, five or six correct guesses. Therefore,
p(pass) =
(6
4
)
+
(6
5
)
+
(6
6
)
64
=
15 + 6 + 1
64
≈ 34%
The expected score from a single question is 0.5, as there is no
penalty for errors. For six questions the expected value is 6 ·0.5 = 3
11
Example
You face a quiz consisting of six true/false questions, and your
plan is to guess the answer to each question (randomly, with
probability 0.5 of being right). There are no negative marks, and
answering four or more questions correctly suffices to pass.
What is the probability of passing and what is the expected score?
To pass you would need four, five or six correct guesses. Therefore,
p(pass) =
(6
4
)
+
(6
5
)
+
(6
6
)
64
=
15 + 6 + 1
64
≈ 34%
The expected score from a single question is 0.5, as there is no
penalty for errors. For six questions the expected value is 6 ·0.5 = 3
12
Exercise
9.3.7
An urn has m + n = 10 marbles, m ≥ 0 red and n ≥ 0 blue.
7 marbles selected at random without replacement.
What is the expected number of red marbles drawn?(m
0
)(n
7
)(10
7
) · 0 + (m1)(n6)(10
7
) · 1 + (m2)(n5)(10
7
) · 2 + . . .+ (m7)(n0)(10
7
) · 7
e.g. (5
2
)(5
5
)(10
7
) · 2 + (53)(54)(10
7
) · 3 + (54)(53)(10
7
) · 4 + (55)(52)(10
7
) · 5
=
10
120
· 2 + 50
120
· 3 + 50
120
· 4 + 10
120
· 5 = 420
120
= 3.5
13
Exercise
9.3.7
An urn has m + n = 10 marbles, m ≥ 0 red and n ≥ 0 blue.
7 marbles selected at random without replacement.
What is the expected number of red marbles drawn?(m
0
)(n
7
)(10
7
) · 0 + (m1)(n6)(10
7
) · 1 + (m2)(n5)(10
7
) · 2 + . . .+ (m7)(n0)(10
7
) · 7
e.g. (5
2
)(5
5
)(10
7
) · 2 + (53)(54)(10
7
) · 3 + (54)(53)(10
7
) · 4 + (55)(52)(10
7
) · 5
=
10
120
· 2 + 50
120
· 3 + 50
120
· 4 + 10
120
· 5 = 420
120
= 3.5
14
Example
Find the average waiting time for the first head, with no upper
bound on the ‘duration’ (one allows for all possible sequences of
tosses, regardless of how many times tails occur initially).
A = E (Xw ) =
∑∞
k=1 k · P(Xw = k) =
∑∞
k=1 k
1
2k
= 1
21
+ 2
22
+ 3
23
+ . . .
This can be evaluated by breaking the sum into a sequence of
geometric progressions
1
2
+
2
22
+
3
23
+ . . .
=
(
1
2
+
1
22
+
1
23
+ . . .
)
+
(
1
22
+
1
23
+ . . .
)
+
(
1
23
+ . . .
)
+ . . .
= 1 +
1
2
+
1
22
+ . . . = 2
15
Example
Find the average waiting time for the first head, with no upper
bound on the ‘duration’ (one allows for all possible sequences of
tosses, regardless of how many times tails occur initially).

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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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 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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