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COMP9020 Random Variables

创建日期：2024-05-25 16:24:13

所属分类：计算机computer science

标签： 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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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 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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 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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 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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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