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STAT 131 Statistics

Creation date：2024-06-01 14:43:24

Belonging category：统计学statistics

Tag STAT 131

Statistics

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STAT 131: Quiz 7 140 total points
Name:
Someone offers you the possibility to play a gambling game with the following rules. First,
you decide how much money you’re willing to put at risk in this game: this amount — let’s
call it A > 0 — is referred to as your stake (all the monetary quantities are in dollars in this
problem); think of A as a fixed known positive real number in what follows. Having chosen
your stake, you’re allowed to bet any amount 0 ≤ B ≤ A (thus, as a decision problem,
your possible actions in this situation correspond to values of B). If you win the bet, which
occurs with probability 0 < p < 1, your stake becomes (A + B); if you lose, it becomes
(A−B), and losing (of course) occurs with probability (1−p); and (crucially) p is known
to you. Let X denote the value of your stake after the gamble has occurred. The point
of this problem is to explore your optimal betting strategy, assuming the Principle of
Maximization of Expected Utility (MEU), for two different reasonable-looking utility
functions, to see which one leads to more sensible behavior.
(a) Write out the probability mass function (PMF) for X. 5 points
(b) 20 points for this part of the problem First let’s suppose that for you, utility
coincides with money, i.e., U1(x) = x.
(i) Work out your expected utility E[U1(X)] as a function of A,B and p. 5 points
1
(ii) Holding A > 0 and 0 < p < 1 constant, what type of function of B is this?
Explain briefly. 5 points
(iii) Using plotting software (or making freehand sketches) and including any result-
ing PDF files in your solution document, graph the expected utility function in
(i) for 0 ≤ B ≤ A in the three separate cases (p < 1
2
), (p = 1
2
), (p > 1
2
); you can
use A = 10 and p = (0.1, 0.5, 0.8) as examples. 10 points
(c) 35 points for this part of the problem Now let’s maximizeE[U1(X)] as a func-
tion of B.
(i) Compute the first partial derivative of E[U1(X)] as a function of B, and try
setting it equal to 0 and solving for B. With reference to your graphs in (b),
briefly explain why this standard calculus approach to maximizing a function
won’t work in this problem. 10 points
2
(ii) By considering the three different cases for p mentioned above, and with par-
ticular reference to your sketches in (b), briefly explain why B∗1 , the optimal B
under U1, is as follows:
B∗1 =

0 (don’t bet) if p < 1
2(
bet any number
between 0 and A
)
p = 1
2
A (bet it all) p > 1
2
 (1)
10 points
(iii) Identify one feature of this betting strategy that seems reasonable, and two
features that seem unreasonable, in all three cases as a function of p; explain
briefly. 15 points
(d) 50 points for this part of the problem Next, let’s suppose instead that you
use Daniel Bernoulli’s utility function U2(x) = 1 + log(x).
(i) Work out your expected utilityE[U2(X)] as a function ofA,B and p. 10 points
(ii) Take A = 10 for illustration and get Wolfram Alpha (or some other equivalent
environment) to plot this as a function of B from −5 to A for two different values
of p: 0.4 and 0.7; reproduce these plots in your solutions, either electronically
or by hand-sketching what your software environment showed you. 10 points
3
(iii) Holding A > 0 and 0 < p < 1 constant, what type of function of B is this
(increasing, decreasing, concave, convex)? Explain briefly. 10 points
(iv) Compute the first partial derivative of E[U2(X)] with respect to B, set this
expression to 0, and solve for B. 10 points
(v) In many problems the computation in (iv) would yield the optimal B∗ (the value
of B that maximizes the expected utility), but the situation is a bit more subtle
here. Identify the range of values of 0 < p < 1 for which the B∗ you obtained
by differentiation cannot be the solution to this optimality problem, and briefly
explain why. 10 points
4
(e) 20 points for this part of the problem Specifying the optimal B with the log-
arithmic utility function.
(i) By examining the two different cases (p ≤ 1
2
), (p > 1
2
), briefly explain why B∗2 ,
the optimal B under U2, is as follows:
B∗2 =

0 (don’t bet) if p ≤ 1
2
2
(
p− 1
2
)
A
(bet an amount
between 0 and A
in proportion to
how much bigger
p is than 1
2
)
 p >
1
2

, (2)
10 points
(ii) Identify two features of this betting strategy that seem eminently reasonable;
explain briefly. 10 points
5
(f) Is it fair to describe one of the two betting strategies above, based on the two different
utility functions, as more risk-seeking than the other one? If so, which is which? If
not, why not? Explain briefly. 10 points

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MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 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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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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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