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FIT5197 Assignment
Creation date：2024-07-16 16:51:13
Belonging category：计算机computer science
Tag FIT5197
Assignment
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FIT5197  Assignment 1 (25 marks)
Contents
1 Details 2
2 Probabilities in Cards (2 marks) 3
2.1 A special flush (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 No repeats (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 PDF and Expectations (3 Marks) 3
3.1 Plot (1/2 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Mean (1/2 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Variance (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.4 Skewness (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Distributions (2 marks) 4
4.1 Model (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2 Checking (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5 Entropy (3 Marks) 4
5.1 Conditional probabilities (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.2 Entropies (1 marks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.3 Coding (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6 Maximum likelihood estimation of parameters (3 marks) 5
6.1 Model (1 mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6.2 Maximum likelihood fitting (2 marks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7 Central Limit Theorem (7 marks) 6
7.1 Sampling distribution (2 marks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7.2 Simulation (2 marks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7.3 Plotting normality (3 marks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Submission due date: by 11:59pm on Friday 12 April 2019 (end of Week 6)
11 Details
Marks
This assignment contains 6 questions. There are 25 marks in total for the assignment and it counts for 25%
of your overall grade for the unit. Also, 3 of the 25 marks are awarded for code quality and 2 of the marks
awarded for presentation of results, for instance use of plots. That leaves 20 marks for individual answers.
You must show all working, so we can see how you obtain your answer. Marks are assigned for working as
well as the correct answer.
Your solutions
Please put your name or student number on the first page of your solutions. Do not copy questions in your
solutions. Only include question numbers. If you use any external resources for developing your results,
please remember to provide the link to the source.
If an extension has been given then submission after the due date is allowed with no penalty being
incurred. If no extension has been given then assignments submitted after the due date, there will be penalised
5% per day up to a maximum of 10 days late.
Submitting your assignment on Moodle
Please submit your assignment through Moodle via upload a Word or PDF document as well as R markdown
you used to generate results.
If you choose to use R markdown file to directly knit Word/PDF document, you would need to type in
Latex equations for Question 1,2 and 5. Find more information about using latex in R markdown files
here. You may also find the R markdown cheatsheet useful.
You can also work with Word and R markdown separately. In this case you would need to type your
answers in Word and also copy R code (using the format: Courier New), results and figures to the
Word document.
We will mark your submission mostly using your Word/PDF document. However, you need to make sure
your R markdown file is executable in case we need to check your code.
Code quality marks
Your R code will be reviewed for conciseness, efficiency, explainability and quality. Inline documentation, for
instance, should demarcate key sections and explain more obtuse operations, but shouldn’t be over verbose.
Out of the 25 marks, 3 will be awarded for code quality.
Presentation marks
Your presentation of results using R will be reviewed. How well do you use plots or other means of ordering
and conveying results. Out of the 25 marks, 2 will be awarded for presentation using R.
22 Probabilities in Cards (2 marks)
Have a regular deck of cards with no jokers (13 cards per suit, 4 suits) giving 52 cards. Suppose we draw a
5 card hand, so 5 cards without replacement. For each answer write out the full calculation in R to show
working.
Note there are 52!
47! different 5 card hands if ordering of the draw is considered, and each is equally likely. If
ordering of the draw is ignored, there are
different 5 card hands.
2.1 A special flush (1 mark)
What is the probability of getting a royal flush but where the cards ordered by rank have alternate color?
That is, order the cards as 10,J,Q,K,A and then check to see they have alternate colour. Note in a proper
royal flush, it is all the one suit, but we have changed that to alternate colour. So, for example “red 10,
black J, red Q, black K, red A” is OK but “red J, black 10, red Q, black K, red A” is not OK because once
reordered in rank the alternating colour no longer holds. Note the order in which they are drawn from the
pack is not considered.
HINT: This event is defined ignoring the order of the draw, so count out the number of such hands (ignoring
the order of the draw), and divide by
2.2 No repeats (1 mark)
What is the probability that in the sequence of cards, as they are drawn, no rank occurs twice in a row?
So ignoring the suit, the following are allowed: A, 10, 4, J, 10 or A, 10, A, 4, A, but the following are not
allowed: A, A, 10, 4, A (A repeated in positions 1 and 2), A, 4, 10, 10, J (10 repeated in positions 3 and 4).
HINT: This event is defined using the order of the draw, so count out the number of such hands, and divide
by 52!/47!.
3 PDF and Expectations (3 Marks)
Let X have the PDF given by a function with a different negative and positive part.
f(x) = 12
You can use Wolfram Alpha to do the definite integrals, for instance
https://www.wolframalpha.com/input/?i=integral+(1-x)%5E3+from+0+to+1
3.1 Plot (1/2 mark)
Draw the plot in R.
33.2 Mean (1/2 mark)
Find E(X). Why is it not zero?
3.3 Variance (1 mark)
Find variance, V ar(X).
3.4 Skewness (1 mark)
Find skewness, using the formula in the lecture notes. Interpret the value.
4 Distributions (2 marks)
One study has evaluated a number of leukaemia records in a rural area. The population of the area was
35,000. In a year there were 16 leukaemia cases identified, of which 4 where not local residents but tourists
or new immigrants (of which there are not many). In a general population, the annual rate of leukaemia is
typically about one in 10,000.
4.1 Model (1 mark)
Describe the model you recommend to use for the counts, and estimate the parameters using suitable point
estimates.
4.2 Checking (1 mark)
Also, consider the hypothesis, “the annual rate of leukaemia in the area is 1/10,000?” Assume this is the rate
for the residents only. Plot the distribution over counts under this hypothesis. Where does your data lie, and
do you think it is consistent with the hypothesis?
5 Entropy (3 Marks)
In this question, we will use a modified version of the Titanic dataset from the Kaggle competition, Titanic:
Machine Learning from Disaster? The dataset includes information about passenger characteristics as well as
whether they survived from the disaster.
Import the Titanic data using the following R code:
df <- read.csv("Titanic.csv",header=TRUE, sep=",")
Now Survived is Boolean so convert to a truth value with:
df[['Survived']] <- df[['Survived']]==1
45.1 Conditional probabilities (1 mark)
Compute tables for the frequency estimates of P(Survived), P(Survived|P class = val) and
P(Survived|Gender = val), for different vals. Do the computation in R. But its OK to present
the final table as a separate Word table (since it might be hard to layout in R). What does this tell you
about survival?
5.2 Entropies (1 marks)
Calculate the entropy (log2()) of Survived, H(Survived) and the conditional entropy of Survived given
P class, H(Survived|P class), and of Survived given Gender, H(Survived|Gender). Do not use an entropy
function but write the code yourself. Use R functions table() and prop.table() to gather stats and form
probabilities from the data frame. What do these three entropies tell you about Survived?
5.3 Coding (1 mark)
Consider the joint space (Survived, P class) which has six outcomes, (T rue, 1), (T rue, 2), (T rue, 3), (F alse, 1),
(F alse, 2), (F alse, 3). Develop an efficient binary prefix code to transmit these outcomes. Would it be
adequate to just provide the codelengths, or is a code needed too? Justify your answer.
6 Maximum likelihood estimation of parameters (3 marks)
One of the central problems of sensory neuroscience is to separate the recordings of background physiological
processes that are irrelevant (noise), from neural responses that are of experimental interest (signal). This
is by no means an easy task, as the signals that neurons produce when they fire are extremely weak and
more random. It is therefore of particular interest to examine the randomness of neuro signals as this allows
researchers to study the brain at a cellular level.
Let’s assume that we have conducted one experiment and recorded the spike signals from one particular
neuron for a duration of 15 seconds. After some data processing, we can obtain spike signals with data given
by a time in seconds and a spike size, similar to the following data and in Figure 1.
n <- 30
times <- c(0.8670763, 1.2550631, 1.3463051, 2.6999393, 3.5238785, 4.8215638, 4.8502006,
5.2372364, 5.3201143, 6.2835730, 7.6961491, 8.0164785, 8.6279902, 9.1390150,
9.5136710, 9.9207854, 9.9795974, 10.0242579, 10.1622076, 10.5968354,
11.6766725, 12.3441424, 12.7731282, 12.8911034, 13.0458095, 13.4280567,
14.2443711, 14.4219672, 14.7461019, 14.7726211)
spikes <- c(0.220136914, 1.252061356, 0.943525370, 0.907732787, 1.157388806, 0.342485956,
0.291760012, 0.556866189, 0.738992636, 0.690779640, 0.425849738, 0.876344116,
1.248761245, 0.697514552, 0.174445203, 1.376500202, 0.731507303, 0.483036515,
0.650835440, 1.106788259, 0.587840538, 0.978983532, 1.179754064, 0.941462421,
0.749840071, 0.005994156, 0.664525928, 0.816033621, 0.483828371, 0.524253461)
6.1 Model (1 mark)
Let us assume that the rate of signals remains constant over time, and the size of each signal is independent
of time too. If the rate of the signals remains constant over time, which distribution would most suit to
model the probability distribution for the number of spike signals over 15 seconds? Why? Briefly answer this
question in a sentence or 2. Also, while we don’t know enough to suggest a distribution for spike sizes, but
what properties should it have?
5Figure 1: Spike data.
6.2 Maximum likelihood fitting (2 marks)
Using the model above, what is the log-likelihood function for number of spike signals for the period of
experiment time, and what is the maximum likelihood estimate for its parameters?
You’re told that a candidate distribution for spike sizes is the Weibull with shape given by 0.7 and unknown
scale, between 0.5 and 2. This is supported in R using the [dpqr]weibull() functions. One can do maximum
likelihood fitting using the Weibull density on the unknown parameter. Use the optimize() function for that,
so something like
‘optimize(fn, c(minvalue,maxvalue), maximum = TRUE, tol = .Machine$double.eps?0.25)’
7 Central Limit Theorem (7 marks)
Assume that we draw random integers from a Poisson distribution with rate one of λ1 = 1, λ2 = 5, or λ3 = 20.
7.1 Sampling distribution (2 marks)
According to Central Limit Theorem what is the limiting distribution for the sample mean, for the three
rates λ1, λ2, λ3, when we have sample size of 10, 100, 1000 and 10000? Give the theory then compute the
parameter values in R.
Bonus question for HD students giving bonus 1 mark (added to final mark for Assignment 1 only if the final
mark is 24 or less): what is the limiting distribution for the sample variance? This is not really a solvable
problem, so approximate it for just λ3 = 20.
67.2 Simulation (2 marks)
Experimentally justify the result in the CLT that says the sample mean has a mean given by the population
mean and a variance given by the population variance divided by sample size. See the CLT Theorem in
Lecture 4. Use simulation given sample a size of 10, 100 and 1000. For each given sample size use 50000
simulations to compute samples and their means. From these means compute the mean and variance of the
sample means, and discuss how results reflect the CLT. Plot the results (3 sample sizes and 3 rates with
mean and SD) to demonstrate any effects you want to discuss.
7.3 Plotting normality (3 marks)
When rate λ1 = 1 and λ2 = 5 and sample size is 10 or 100, obtain the z scores of the sampling means (from
50000 simulations). Plot their distributions in a histogram with the theoretical Gaussian curve overlaid.
Note for sample size 100, the plots overlay very nicely. But what happens with sample size 10? Explain the
differences between the four plots.
For each simulation: the z score of the mean can be calculated as:
where Xˉ is the mean of the sample, μ is the population mean and σ is the population SD.
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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 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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 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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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 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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