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### stat400lec25

Course: STAT 400, Fall 2008
School: University of Illinois,...
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400 Statistics Section 5.3 Distribution of sums of the independent random variables Review If X1, X2, ..., Xn are n independent random variables with respective 2 2 2 means 1, 2, ..., n and variances 1 , 2 , ..., n , then the mean and n variance of Y= 1 ai Xi , Y = 1 ai i , n 2 Y = 1 ai2 i2 n If X1, X2, ..., Xn are n independent random variables with respective moment-generating function MXi (t), then the...

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Coursehero >> Illinois >> University of Illinois, Urbana Champaign >> STAT 400

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400 Statistics Section 5.3 Distribution of sums of the independent random variables Review If X1, X2, ..., Xn are n independent random variables with respective 2 2 2 means 1, 2, ..., n and variances 1 , 2 , ..., n , then the mean and n variance of Y= 1 ai Xi , Y = 1 ai i , n 2 Y = 1 ai2 i2 n If X1, X2, ..., Xn are n independent random variables with respective moment-generating function MXi (t), then the moment-generating n function of Y= 1 ai Xi is M Y ( t ) = M X i ( ai t ) , 1 n Theorem 5.3-1 If X1, X2, ..., Xn are observations of a random sample of size n from a normal distribution N( , 2), then distribution of the sample mean X is Ping Ma Lecture 25 -1- Review Let the distribution of independent random variables X1, X2, ..., Xn be 2(r1), 2(r2),..., 2(rn), respectively, then distribution of Y= X1+X2+ ... +Xn is 2(r1+r2+...+rn) Theorem 5.3-2 If Z1, Z2, ..., Zn are observations of a random sample of size n from a standard normal distribution N(0,1), , then W= Z12+Z22+ ... +Zn2~ 2(n) Ping Ma Lecture 25 -2- Theorem 5.3-3 If X1, X2, ..., Xn are observations of a random sample of size n from a normal distribution N( , 2), then we have (a) X and S2 are independent (b) ( n - 1) S 2 is 2(n-1) 2 Ping Ma Lecture 25 -3- If X1, X2, ..., Xn are n independent norma;l random variables with 2 2 2 respective means 1, 2, ..., n and variances 1 , 2 , ..., n , then the n mean and variance of Y= 1 ai Xi , Then Y~N( Y , Y ) 2 Where Y = 1 ai , n 2 i Y = 1 ai2 i2 n Ping Ma Lecture 25 -4- Student's t-distribution Let Z`N(0,1) and U is a chi-square random variable with r degrees of freedom, Z and U are independent T= Z U /r is called t distribution Important special case: Let Z1 ~ N(0,1), Z2~N(0,1) and Z1 and Z2 are independent, then T= Z1 ~ t(1) Z2 Properties of t distribution (1) symmetric and unimodal (2) Bell-shaped curve has heavier tail than normal distribution 100(1-) percentile of t distribution with r degree of freedom t(r) So we have P( T t(r) )= Table VI. Ping Ma Lecture 25 -5- Example: Let T have a t distribution with 14 degree of freedom. Find the constant c such that P(|T|<c)=0.90 Ping Ma Lecture 25 -6- If X1, X2, ..., Xn are observations of a random sample of size n from a normal distribution N( , 2), then we have (a) X is N( , 2/n) (b) X - is N(0,1) / n ( n - 1) S 2 is 2(n-1) 2 (c) ...

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University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.4 Central limit theorem Central Limit Theorem (CLT) If X1, X2, ., Xn are observations of a random sample of size n from a distribution with mean and variance 2 , Then we have W=X - X i - n = N(0,1) as / n nnIn anoth
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.5 Central limit theorem Central Limit Theorem (CLT) If X1, X2, ., Xn are observations of a random sample of size n from a distribution with mean and variance 2 , Then we have W=X - X i - n = N(0,1) as / n nnIn anoth
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.6 Bivariate normal distribution Two random variables X and Y have a bivariate normal distribution with 2 means X and Y , variances X and Y2 , correlation coefficient .2 X X X ~ , N Y Y X Y X Y
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 29 Review Continuous distribution: Probability density function Properties of p.d.f f(x): (a) f(x)&gt;0; (b) f ( x )dx = 1 ; (c) P(a&lt;X&lt;b)= a f ( x )dx Cumulative distribution function (c.d.f) F(x)= P( X x ) = F'(x)=f(x) Expected
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Chapter 6 Estimation Maximum likelihood estimates Example: If X1, X2, ., X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find Blah blah blah How do we know that the mean is 50 and the variance i
University of Illinois, Urbana Champaign - STAT - 400
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University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Confidence intervals for means The Statistical End Goal: draw conclusions about the data we have collected and analyzed. Every time we estimate using statistic X , we will never get the same answer. Due to sampling variability Need t
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.5 Confidence Intervals for Proportions Binomial distribution Let Y ~ b(n,p) Objective: construct confidence interval for pY - np Y /n- pW= np(1 - p ) = p(1 - p ) / n N(0,1) &quot;sufficiently large&quot;: np 5 and n(1-p) 5Ping
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.7 Sample Size Calculation Review: Confidence Interval Form: estimate margin of error n s x t / 2 * nx z / 2 *Y Y / n (1 - Y / n ) z / 2 n nThe length of the interval = 2*margin of error Tow factors associated with w
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests of Statistical Hypothesis Confidence Intervals: Creates an interval where we think the true parameter we are estimating will fall with a certain probability or level of confidence. Serve the purpose when the goal is to estimate
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests of Statistical Hypotheses Simple hypothesis Composite HypothesisTypes of Error Type I Error: If we reject Ho when Ho is true Type II Error: If we fail to reject Ho when Ho is false Ho True Reject Ho Type I error Ha True Correct
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests about Proportions Recall:^ Proportions: ( p = ) Y nCount the number of successes and take into account the sample Normal Approximation of proportions Draw a random sample of size n from a large population with p = P ^ (Success
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Comparing Two Means and Two Variances Two Sample Problems: Compare the responses in two groups Each group is an individual sample from a distinct population The responses from either group are independent of each other When a two-sa
University of Illinois, Urbana Champaign - STAT - 400
Final Review We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Estimator: The function of X1, X2,., Xn used to estimate , say the st
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Virginia Tech - CS - 6104
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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Kentucky - CHE - 226
Department of ChemistryUniversity of KentuckyEXPERIMENT 5 Molecular Absorption Spectroscopy: Determination of Iron with 1,10-PhenanthrolineUNKNOWN Submit a clean, labeled 100-mL volumetric flask to the instructor so that your unknown iron solut
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Maryland - CMSC - 351
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Sveriges lantbruksuniversitet - LING - 220
Tree structures from the Nov 6-8 tutorials: Question 3 a) Det the NP N zoo b) Qual always VP V tryc) Deg soAP A wittyd)VP V passQual perhapse) Deg less g) Deg very i) Qual neverAP A bleak AP A competent VP V surrenderf) Det this h) De
Sveriges lantbruksuniversitet - LING - 220
11-6-00 Note: lines are missing from these notes. - Announcements - Assignment 3 is due in your tutorial section this week, November 6, 7, or 8. - Syntax, phrase structure Ahead - NO CLASS ON MONDAY REMEMBRANCE DAY - Wednesday: syntax, continued -Sy
Sveriges lantbruksuniversitet - LING - 220
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Sveriges lantbruksuniversitet - LING - 220
Tree structures from the Nov 13-15 tutorials: Question 5 a) SNP Det Those N guestsI Aux shouldVP V leaveb)S I [+pst] VPNP N Maria c) S Qual never V ate Det aNP N brownieNP Det That N shelfI Aux willVP V falld)S I [+pst] NP VP
Sveriges lantbruksuniversitet - LING - 220
11-18-00 - Announcements - Assignment 4 is due in your tutorial this week (Nov. 20, 21, 22). Assignment 5 will be handed out in your tutorial and due the following week. - cross-linguistic variation - Functional categories of syntactic structures - P
Sveriges lantbruksuniversitet - LING - 220
10-02-02 These notes are essentially the same as those posted before class. Please note that various symbols and lines are missing. - Announcements: - Dr. Mellow will not have office hours on Thursday Oct 3: He will be at a conference in Toronto. - T
Sveriges lantbruksuniversitet - LING - 220
10-07-02 - Announcements - Review (minimal pairs, complementary distribution, solving phonology problems) - reading of words from Assignment 2 - solving phonology problems Ahead Phonology continued, problem solving (questions 3, 4, and 5 on pp. 113-1
Sveriges lantbruksuniversitet - LING - 220
Tutorial Activities for Oct 30- Nov1 1. Draw tree structures for the following complex words: These trees do not have lines, but I have tried to show the hierarchical internal structure with vertical spacing and intendation. N V A modern Af iz Af ati
Sveriges lantbruksuniversitet - LING - 220
9-30-02 - Announcements: - If you missed your tutorial section due to illness, you can pick up a copy of Assignment 1 from Dr. Mellow during his office hours (Mon 10:30-12) - For those students who have learned a different set of phonetic symbols, pl
Sveriges lantbruksuniversitet - LING - 220
On this page, I list 22 more words that you can practice transcribing. On the next page, I provide transcriptions using the symbols we are working with in our course. rich reach ridge rote crude crowd cried wrote fudge kite action pizza rot rat rate
BYU - JOB - 721
13548(DYRK1A)