Western Washington - BIO - 348

Bio 348: Skeletal System GLOSSARY OF TERMS: SKELETAL SYSTEMTerm: condyleDefinition (with one example): a rounded process that articulates with another bone eg. occipital condyle a narrow, ridge-like projection; eg. iliac crest a projection situated abov

Western Washington - BIO - 348

Light is transmitted through: Cornea Aqueous Humor Retina: Through Ganglion cells Through Bipolar Cell LayerLensVitreous HumorStimulating Rods and Cones (Photoreceptors) Absorbed at Pigmented Epithelium layer and Choroid layer (outside Retina) (Stimula

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 1Introduction to StatisticsFor handing in on 17 January 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feedba

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 2Introduction to StatisticsFor handing in on 2425 January 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feed

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 3Introduction to StatisticsFor handing in on 31 January 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feedba

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 4Introduction to StatisticsFor handing in on 7 February 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feedba

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 5Introduction to StatisticsFor handing in on 1415 February 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Fee

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 6Introduction to StatisticsFor handing in on 28 February 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feedb

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 7Introduction to StatisticsFor handing in on 67 March 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feedback

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Assignment 8Introduction to StatisticsFor handing in on 13 March 2012You should attempt all of these questions, as they are designed to help you to learn and understand the material in the course. The `Feedback

Queen Mary University of London - MTH - 4106

B. Sc. Examination by course unit 2010 MTH4106 Introduction to StatisticsDuration: 2 hours Date and time: 6 May 2010, 1430h1630hApart from this page, you are not permitted to read the contents of this question paper until instructed to do so by an invig

Queen Mary University of London - MTH - 4106

B. Sc. Examination by course unit 2011 MTH4106 Introduction to StatisticsDuration: 2 hours Date and time: 4 May 2011, 1430h1630hApart from this page, you are not permitted to read the contents of this question paper until instructed to do so by an invig

Queen Mary University of London - MTH - 4106

MTH4106Notes 1 Exploratory data analysisIntroduction to StatisticsSpring 2012In Probability, we start with a probability distribution for some random variable X, and deduce things like the expectation E(X) or P(X 80). In Statistics, we start with some

Queen Mary University of London - MTH - 4106

MTH4106Notes 3 Discrete random variablesSome revisionIntroduction to StatisticsSpring 2012If X is a discrete random variable then X may take finitely many values x1 < x2 < < xn or infinitely many values cfw_xi : i Z so long as no two are too close to

Queen Mary University of London - MTH - 4106

MTH4106Notes 4 What is Statistics about?Introduction to StatisticsSpring 2012We collect data, then analyse the data, and then interpret the results, to find out about real-world phenomena. There is aways variability in the data: we need to extract mea

Queen Mary University of London - MTH - 4106

MTH4106Notes 5 Estimating a ProportionIntroduction to StatisticsSpring 2012Suppose that there is a population of N items, of which M have Type A and N - M have Type B. Put M p= = proportion having Type A. N We take a random sample of size n. Let X be

Queen Mary University of London - MTH - 4106

MTH4106Notes 6Introduction to StatisticsSpring 2012Testing Hypotheses about a ProportionExample Pete's Pizza Palace offers a choice of three toppings. Pete has noticed that rather few customers ask for anchovy topping. He thinks that if fewer than 1/

Queen Mary University of London - MTH - 4106

MTH4106Notes 7 Continuous random variablesIntroduction to StatisticsSpring 2012If X is a random variable (abbreviated to r.v.) then its cumulative distribution function (abbreviated to c.d.f.) F is defined by F(x) = P(X x) for x in R.We write FX (x)

Queen Mary University of London - MTH - 4106

MTH4106Notes 8 Two or more random variablesTwo continuous random variablesIntroduction to StatisticsSpring 2012If X and Y are continuous random variables defined on the same sample space, they have a joint probability density function fX,Y (x, y) suc

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 1Introduction to Statistics10 January 2012In this practical you will be introduced to the statistical computing package called Minitab. You will use this package throughout this module, and also in so

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 2Introduction to Statistics17 January 2012If you didn't do so last week, please complete Practical 1 before going on to do the following. Remember to move to the Mathematics department > Statistics I

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 3Introduction to Statistics24 January 2012If you didn't do so last week, please complete Practicals 1 and 2 before going on to do the following. Today we will see how to use Minitab to plot and summar

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 4Introduction to Statistics31 January 2012Today we will use Minitab's inbuilt information about the most important discrete random variables. We will learn how to list the values of the probability ma

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 5Introduction to Statistics7 February 2012Today we will use Minitab's inbuilt information about the Binomial distribution to calculate and plot power curves for hypothesis tests about a proportion. Be

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 6Introduction to Statistics14 February 20121 (Plotting a probability mass function as a histogram) In Practical 4 we plotted the probability mass function of several discrete random variables. Sometim

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 7Today's practical does two different things: plotting the probability density function of some normal distributions for you to insert in your lecture notes (this part continues from Practical 6); plott

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 8Today we will see how Minitab simulates data from standard distributions, become more familiar with covariance and the correlation coefficient, verify some theorems from lectures in particular cases. 1

Cal Poly Pomona - ECE - 114

How To Add Program Run Data to Listing 1. After the program runs, right click mouse on top bar of the black(DOS) screen, select edit, select mark see below.2. A white cursor appears in upper left hand corner, with mouse drag it so that all of the run dat

Queen Mary University of London - MTH - 4106

QUEEN MARY, UNIVERSITY OF LONDON MTH 4106Practical 9Today we will draw graphs to show how well the normal distribution approximates the binomial and Poisson distributions, use an example to demonstrate the Law of Large Numbers. You will be asked to incl

Queen Mary University of London - MTH - 4106

Queen Mary University of London - MTH - 4106

Queen Mary University of London - MTH - 4106

Queen Mary University of London - MTH - 4106

Queen Mary University of London - MTH - 4106

Queen Mary University of London - MTH - 4106

Ill. Chicago - STAT - 411

Stat 411 Homework 01Due: Friday 01/201. Let cfw_Xn : n 1 be a sequence of positive independent random variables with E(Xn ) = c (0, 1) for each n. Let Yn = X1 X2 Xn , the product of the Xi 's. Use Markov's inequality to prove that Yn 0 in probability. 2

Ill. Chicago - STAT - 411

Stat 411 Homework 01Solutions1. Since the Xi 's are independent, E(Yn ) = E(X1 ) E(Xn ) = cn . By Markov's inequality, P(Yn > ) -1 E(Yn ) = -1 cn 0, n , since c (0, 1). Since > 0 is arbitrary, Yn 0. 2. (a) The maximum of a list of numbers less than x if

Ill. Chicago - STAT - 411

Stat 411 Homework 02Due: Friday 01/271. Let X1 , . . . , Xn be iid Unif(0, ), where > 0 is unknown. ^ ^ (a) Let n = X(n) , the sample maximum. Find the CDF of n . (Hint: This is similar to Problem #2 on Homework 01.) ^ ^ (b) Show that n is a consistent

Ill. Chicago - STAT - 411

Stat 411 Homework 02Solutions1. (a) The CDF of Unif(0, ) is F (x) = x/, if x (0, ) and the obvious modifications otherwise. Because X1 , . . . , Xn are iid Unif(0, ), the CDF of the ^ ^ maximum n = X(n) is simply Fn, (t) = P (n t) = (t/)n , for t (0, ).

Ill. Chicago - STAT - 411

Stat 411 Homework 031. Problem 6.1.3 on pages 317318.Due: Friday 02/032. Problem 6.1.6 on page 318. [Hint: Refer back to Problem 6.1.3(c).] 3. Problem 6.1.9 on page 318. [Hint: Refer back to Problem 6.1.3(a).] 4. Problem 6.1.11 on page 319. 5. Let be a

Ill. Chicago - STAT - 411

Stat 411 Homework 031. Problem 6.1.3.Solutions(a) Let X1 , . . . , Xn be iid with PMF f (x) = e- x /x!, x = 0, 1, 2, . . . (a Poisson distribution). Then the likelihood function isnLx () =i=1e- xi = const e-n x1 +xn . xi !Taking log and then deri

Ill. Chicago - STAT - 411

Stat 411 Homework 04Due: Wednesday 02/08Undergraduates may solve the "Graduate only" problem(s) for possible extra credit. 1. Problem 6.2.7 on page 331. 2. Problem 6.2.9 on page 332. (Hint: Find expected value using integration-by-parts.) 3. Theorem 6.2

Ill. Chicago - STAT - 411

Stat 411 Homework 04Solutions1. Problem 6.2.7 in the text. The PDF for the Gamma(4, ) distribution is f (x) = 1 3 -x/ xe , 64 x > 0, > 0.(a) For the Fisher information, we first need second derivative of log-PDF: x 4 2 2 2x = 2 - 3. log f (x) = 2 const

Ill. Chicago - STAT - 411

Stat 411 Homework 05Due: Wednesday 02/22Undergraduates may solve the "Graduate only" problem(s) for possible extra credit. 1. Let X1 , . . . , Xn be an iid sample from a log-normal distribution with PDF 1 2 f (x) = e-(log x-1 ) /22 , x 22 (a) Find the M

Ill. Chicago - STAT - 411

Stat 411 Homework 051. (a) The likelihood function looks likenSolutionsL(1 , 2 ) =i=1f (Xi ) 2-n/2 -(1/22 )en 2 i=1 (log Xi -1 ).Taking a natural logarithm gives (1 , 2 ) = const - 1 n log 2 - 2 22n(log Xi - 1 )2 .i=1Differentiating with re

Ill. Chicago - STAT - 411

Stat 411 Homework 06Due: Wednesday 02/29Undergraduates may solve the "Graduate only" problem(s) for possible extra credit. 1. Let X1 , X2 be iid with PDF f (x) = (1/)e-x/ , x > 0. (a) Let Y1 = X1 and Y2 = X1 +X2 . Find the joint PDF fY1 ,Y2 (y1 , y2 ),

Ill. Chicago - STAT - 411

Stat 411 Homework 06Solutions1. (a) First need to find the joint PDF of Y1 = X1 and Y2 = X1 + X2 . The Jacobian of the (inverse) transformation is 1, so the joint PDF of (Y1 , Y2 ) is given by fY1 ,Y2 (y1 , y2 ) = fX1 ,X2 (y1 , y2 - y1 ) = (1/2 )e-y2 /

Ill. Chicago - STAT - 411

Stat 411 Homework 07Due: Wednesday 03/07Undergraduates may solve the "Graduate only" problem(s) for possible extra credit. 1. Let X1 , . . . , Xn be iid N(, 1). Find the MVUE of = 2 . 2. Problem 7.5.3 on page 392. 3. Problem 7.5.6 on page 393. 4. Proble

Ill. Chicago - STAT - 411

Stat 411 Homework 07Solutions1. The distribution N(, 1) is a regular one-parameter exponential family problem with K(x) = x. Therefore, T = n Xi is a complete sufficient statistic for and, i=1 2 ^ consequently, the MVUE of is X = T /n. It is easy to che

Ill. Chicago - STAT - 411

Stat 411 Homework 08Due: Wednesday 03/14Undergraduates may solve the "Graduate only" problem(s) for possible extra credit.2 1. Let X1 , . . . , Xn N(1 , 2 ), where 2 > 0 is the variance. Find the MVUE of 1 .iid2. Problem 7.7.10 on page 405. (Hint: Us

Ill. Chicago - STAT - 411

Applications of Basu's theoremDennis D Boos; Jacqueline M Hughes-Oliver The American Statistician; Aug 1998; 52, 3; ABI/INFORM Global pg. 218Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.Reproduce

Ill. Chicago - STAT - 411

Stat 411 Lecture Notes Point estimation Ryan Martin Spring 20121IntroductionThe statistical inference problem starts with the identification of a population of interest, about which something is unknown. For example, before introducing a law that home

Ill. Chicago - STAT - 411

Source: http:/www.math.cuhk.edu.hk/~mat2060/mat2060b/Notes/notes3.pdf2005-06 Second Term MAT2060B Supplementary Notes 3 Interchange of Differentiation and Integration1The theme of this course is about various limiting processes. We have learnt the limi

Ill. Chicago - STAT - 411

Stat 411 Lecture Notes Supplement Computation of maximum likelihood estimators Ryan Martin Spring 20121IntroductionSuppose that sample data X1 , . . . , Xn are iid with common distribution having PMF/PDF f (x). The goal is to estimate the unknown para

Ill. Chicago - STAT - 411

Stat 411 Lecture Notes Supplement RaoBlackwell and LehmannScheffe theorems Ryan Martin Spring 20121IntroductionUnbiased estimation is a fundamental development in the theory of statistical inference. Nowadays there is considerably less emphasis on unb

Ill. Chicago - STAT - 411

Stat 411 Lecture Notes Statistics and sampling distributions Ryan Martin Spring 20121IntroductionStatistics is closely related to probability theory, but the two fields have entirely different goals. Recall, from Stat 401, that a typical probability p

UNC - EPID - 600

Principles of Epidemiology for Public Health (EPID600) Introduction to the courseFaculty: Victor J. Schoenbach, PhD Lorraine K. Alexander, PhDhome pageDepartment of Epidemiology Gillings School of Global Public Health University of North Carolina at Ch

UNC - EPID - 600

Principles of Epidemiology for Public Health (EPID600)The population perspectiveVictor J. Schoenbach, PhD home page Department of Epidemiology Gillings School of Global Public Health University of North Carolina at Chapel Hillwww.unc.edu/epid600/1/18/

UNC - EPID - 600

Principles of Epidemiology for Public Health (EPID600)Epidemiologic measures: Incidence & prevalenceVictor J. Schoenbach, PhD home page Department of Epidemiology Gillings School of Global Public Health University of North Carolina at Chapel Hillwww.un

UNC - EPID - 600

Principles of Epidemiology for Public Health (EPID600)Natural history of disease / population screeningVictor J. Schoenbach, PhD home page Department of Epidemiology Gillings School of Global Public Health University of North Carolina at Chapel Hillwww

UNC - EPID - 600

Principles of Epidemiology for Public Health (EPID600)Study designs: Intervention trialsVictor J. Schoenbach, PhD www.unc.edu/~vschoenb Department of Epidemiology Gillings School of Global Public Health University of North Carolina at Chapel Hillwww.un