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MIE 360: Systems Modeling and Simulation
创建日期：2024-06-04 15:24:43
标签： MIE 360
Systems Modeling and Simulation

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MIE 360: Systems Modeling and Simulation 

Tutorial 2: CLT and Monte Carlo Simulation
1. An insurance company serves five large customers. The insurance claim placed by each cus-
tomer in a year is exponentially distributed with mean 200. Assume that the insurance claims
from different customers are independent from each other.
(a) Estimate the probability that the total insurance claim placed by all customers in a year
exceeds 900 by using the CLT.
(b) Use simulation to estimate this probability. First, find the required number of samples
to achieve ±0.01 accuracy with 95% confidence using the approach discussed in the
lecture. Then use the obtained number of samples to get the estimate. Implement
your solution using Python. You might find the list of Python functions at the end
of this document helpful. (Hint: In this question you may use the Python function
numpy.random.exponential(scale=α, size=n) to generate n random exponential numbers
with mean α. Later in the course you will learn how to transform uniform random
numbers into exponential ones.)
(c) Repeat parts a) and b) for 30 customers’ total claims exceeding 5,400.
2. The lifetime of a special type of battery has mean 40 hours and standard deviation 20 hours
(the unknown distribution does not take negative values). A battery is used until it depletes
at which point it is replaced by a new one. Use the central limit theorem to estimate the
number of required batteries, the lifetimes of which are independent, so that the probability
that we obtain a total of 1,100 hours of use is more than 88.5%.
3. Assume that you rolled a die for 10,000 times and the outcomes that you obtained are given
by the data in the attached spreadsheet (“Tutorial 2.xlsx”). You suspect if the die is fair or
not.
Note: You can download the attached “Tutorial 2.ipynb” Jupyter (IPython) Notebook which
contains the code for reading the data and storing it as a numeric array. You need to put the
file (Tutorial 2.xlsx) and Tutorial 2.ipynb in the same folder before executing the code.
(a) Compute a 95% CI for the mean roll of the given die based on the given data, and
compare your CI to the value you should obtain if the die is actually fair. Does the die
look fair?
(b) Compute a 95% CI for the probability that a roll of the given die yields the value 1, and
compare your CI to the value you should obtain if the die is actually fair. Does the die
look fair?
(c) Explain the differences, if any, in your conclusion between parts a and b.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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