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MATH 368 function

创建日期：2024-05-24 15:33:35

所属分类：数学mathematics

标签： MATH 368

function

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MATH 368

Task 1. [5+5 points]
Your group has a special pdf πg for generating a discrete random variable X with values in {1, 2, 3}.
This πg = (π(1), π(2), π(3)) with π(i) = Pr(X = i), i = 1, 2, 3, is obtained as follows: Subtract
your group number g from 559 and express the result as 3-digit decimal number a1 ·102+a2 ·10+a3
with a1, a2, a3 ∈ {0, . . . , 9}. Then,
π(i) := ai/(a1 + a2 + a3), i = 1, 2, 3.
(a) Give a Matlab implementation of a function GenerateRandomNumbers(n,g) that takes a
number n and your group number g as input and returns n random numbers from {1, 2, 3}
following your pdf πg. Your function is not allowed to use any built-in number generators
except for rand.
(b) Plot a histogram of nTrials=10^6 random numbers generated by your function
GenerateRandomNumbers (using your πg). The bins should be centered at 1, 2, and 3.
Task 2. [5+5 points]
Consider the following problem (a version of the so-called Monty Hall problem). A car is randomly
placed behind one of the three doors #1, #2, #3 following your special pdf πg from Task 1
(i.e., the probability that the car is placed behind door #i is π(i), i = 1, 2, 3). The contestant
chooses door #1. If the car is behind door #1, the host chooses door #2 with probability q and
door #3 with probability 1 − q. If the car is not behind door #1, then the host opens whichever
of doors #2 or #3 that the car is NOT behind. The contestant makes now a final choice, sticking
to door #1 or switching to the unopened door. The contestant wins if the car is behind the final
chosen door.
Several strategies are available to the contestant. Relevant are here the following three strategies:
Strategy 1: Stick always to the original choice, door #1.
Strategy 2: Switch always to the unopened door.
Strategy 3: Switch always to door #3 if door #2 is opened, otherwise stick to door #1.
(a) Implement a Matlab function that performs simulations to estimate the probability of win-
ning for each of the three strategies.
(b) Provide a single plot that shows these estimated probabilities of winning, for each of the
three strategies, as a function of q.
Task 3. [8+8+6+8 points]
Alice wants to solve six computational tasks (T1, . . . , T6) on her computer, which is equipped with
five Central Processing Units (CPU1, . . . , CPU5). Each task can be split into a certain number
of subtasks that can be processed in parallel (i.e., on different CPUs). However, each CPU can
Page 1 of 2
process only a specific number of subtasks (otherwise it would overheat).
Taking all these constraints into account, Alice came up with the computation plan shown below.
It is given in form of a binary matrix A with entries aij, i = 1, . . . , 5, j = 1, . . . , 6, where aij = 1
if, and only if, a subtask of Tj is processed on CPUi.
A =
T1 T2 T3 T4 T5 T6

a11 a12 a13 a14 a15 a16 CPU1
a21 a22 a23 a24 a25 a26 CPU2
a31 a32 a33 a34 a35 a36 CPU 3
a41 a42 a43 a44 a45 a46 CPU4
a51 a52 a53 a54 a55 a56 CPU5
=
1 2 3 4 5 6

1 0 1 0 1 1
1 1 0 0 0 1
0 1 1 1 1 1
1 0 1 1 1 1
1 1 1 0 1 1
Alice wonders whether there are other, ‘equivalent’, computation plans. A computation plan, i.e.,
a binary matrix B, is called equivalent to A if in each row and each column B has the same number
of 1s as A. A matrix A′ obtained from A by replacing a 2× 2 submatrix of the type(
1 0
0 1
)
or
(
0 1
1 0
)
by the other is said to be obtained from A by an interchange. It is a classical result (and you can
use it without proof) that B is equivalent to A if, and only if, B can be obtained from A by a
sequence of interchanges.
(a) Write a Matlab function EnumerateAllSolutions(A), which takes Alice’s A as input and
returns, by checking systematically all 5×6 binary matrices, the number of different matrices
that are equivalent to A. How many different matrices are equivalent to A?
(b) Write a Matlab function RunMarkovChain(A), which takes Alice’s A as input and generates
a Markov Chain, based on interchanges, that samples the matrices equivalent to A uniformly
at random. Provide a histogram that demonstrates that your RunMarkovChain(A) samples
the matrices equivalent to A uniformly at random.
(c) Prove that your Markov Chain from (b) is aperiodic, irreducible, and has the uniform distri-
bution as stationary distribution.

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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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