Introduction to Mathematical Statistics
Introduction to Mathematical Statistics STAT 310
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Mrs. Triston Collier
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This 2 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 310 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/205086/stat-310-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.
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Date Created: 09/17/15
STAT 310 DISCUSSION 1 TA Yi Chai Of ce 1335N MSC Phone 2635948 Email chaiyi statwiscedu Webpage httpwwwstatwisceduchaiyi Of ce Hours 11001200pm T and 100200pm Th or by appointment 1 Summary of section 51 and 52 0 Statistics is applied to situation in which we have questions that cannot be answered de nitively7 typically because of variation in data 0 Probability is used to model the variation observed in the data Statistical inference is concerned with using the observed data to help identify the true probability distribution or distributions producing this variation and thus gain insight into the answers to the questions of interest 0 Probability models are used to model uncertainty about future responses 0 We can use the probability distribution to predict a future response or assess whether or not a given value makes sense as a possible future value from the distribution 2 Brief Review 0 WLLNWeak Law of Large Numbers Let X17 X27 be a sequence of independent random variables7 each having the same mean it and each having a nite variance Then 1 n E XiL M n 11 0 CLTCentral Limit Theorem Let X17X27 be a sequence of independent random variables7 each having the same mean it and same nite variance 02 Then ZnLZNO1 3 Introduction to R 0 available statistical softwares on campus Computer Labs R Chadbourne Hall Rm 113 Gordon Commons Rm A2 Showerman Kronshage Lower Level Elizabeth Waters Hall Rm 3414 Animal Science Rms 149 and 150 College Library Rm 2250 Health Sciences Learning Center Rm 2110 Memorial Library Rm 140 Steenbock Library 2nd Floor Union South Lower Level Wendt Library 2nd Floor 0 Homepage of R httpwwwrprojectorg 0 Basic commands of R helpstart start the search engine in R qu39nction get more information on any speci c named function x C123 create a vector named x which has 3 elements 12 and 31 41 Examples 0 Example 1 51111 in the textbook 0 Example 2 Suppose the following data were generated from an N051 distribution by a student However the student forgot which of mean was used so we are uncertain about the correct probability distribution to use to describe the variation in the data Can you suggest a plausible value for 1 Explain your reasoning quot1391quot quot3394quot quot3397quot quot1395quot quot0399quot quot0399quot quot2395quot quot0396quot quot4395quot quot0397quot quot1398quot quot4394quot quot1390quot quot2390quot quot2392quot quot0394quot 1 i 11911765 0 Example 3 Suppose that we want to obtain the distribution of the quantity Y X3 6X 7 3 when X N N0 1 Here we are faced with a form of mathematical uncertainty because it is very dif cult to determine the distribution of Y using mathematical methodsi Propose a computer method for approx imating the distribution function of Y and estimate PY E 0 What is the relevance of statistical methodology to your approach gt x rnorm1000 x 3expx3 gt hist ybreaks40x1imc2020 gt sumygt0 amp ylt5 1 135 V amplt Histogram of y Frequency 2 100 0 Example 4 Suppose that the life length of a machine is known to be distributed as Y 10X where X N Uniform0 l 7 Find the smallest interval containing 95 of the probability for Y 7 Find the probability that the life length of a machine is longer than 5 years 7 Find the expectation of the life length of a machine if it has already been running for 1 year 7 Find PY gt SlY gt1 7 Find the smallest interval containing 95 of the conditional probability