Prerequisites for the lectures taught in the Statistics

Prerequisites for the lectures taught in the Statistics

Prerequisites for the lectures taught in the
Statistics Master program at the
Biostatistics Unit, UZH
Leonhard Held & Kaspar Rufibach
Biostatistics Unit,
Institute of Social and Preventive Medicine,
University of Zurich
Hirschengraben 84, CH-8001 Zurich, Switzerland
http://www.biostat.uzh.ch
February 2009
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Purpose
This document lists the prerequisites for the lectures taught in the Master program in Statistics at
the Biostatistics Unit, Institute for Social and Preventive Medicine, University of Zurich. Lectures
currently taught are
• “Angewandte Mathematische Statistik” (Applied Mathematical Statistics),
• “Bayes Methoden” (Bayesian Statistics),
• “Biostatistische Methoden” (Biostatistical Methods).
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General prerequisites
The lectures taught address an audience that has a sufficient training in
• analysis,
• linear algebra,
• probability, and
• statistics.
Former students with a non-mathematical bachelor found Cramer and Neslehova (2008) helpful.
In addition, we use the software R in lectures and exercises. A basic knowledge of R is assumed.
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Prerequisites in more detail
We provide a list of topics and concepts that we assume in our lectures. The courses “Angewandte
Mathematische Statistik” and “Bayes Methoden” follow Held (2008) closely.
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3.1
Analysis
• basic properties of functions (such as continuity, monotonicity, ...)
• differential calculus,
• integral calculus,
• Landau’s notation,
• Taylor’s Theorem,
All these topics are covered in Heuser (2006a,b). Former students found Thompson and Gardner
(1998) helpful as well.
3.2
Linear algebra
We assume a basic understanding of vector and matrix calculus. More specifically,
• basic operations with matrices,
• concepts like “determinant”, “positive definiteness”, etc.,
• inversion of matrices,
• how to solve a system of linear equations,
• Cholesky factorization of a matrix.
Former students found Jänich (2008) and Anton and Rorres (2005) helpful. We also recommend
Strang (2005).
3.3
Probability
We assume the content of a one-semester, 4/week introductory course in probability. Specific topics
are:
• events, probabilities and conditional probabilities,
• random variables and random vectors,
• univariate density and distribution function,
• multivariate density and distribution function,
• density and distribution function of an i.i.d. sample of observations,
• expectation and variance of a function of a random variable or vector,
• density of a function of a random variable (“change of variables”),
• conditional distributions and conditional expectation,
• Law of large numbers and central limit theorem.
These topics are well covered in Grimmett and Stirzaker (2001, Chapters 1-5). Some of it is discussed
more informally in Fahrmeir et al. (2007, Chapters 4-8). In addition, we assume that students are
familiar with standard distributions used in statistics, as provided in Held (2008, Appendix A.3).
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3.4
Statistics
We assume the content of a one-semester, 4/week introductory course in statistics. Important topics
are:
• “population” and “sample”,
• “descriptive” and “inferential” statistics,
• confidence intervals,
• statistical tests: concepts and standard tests,
These topics are well covered in Fahrmeir et al. (2007, Chapters 1-3 and 9-11).
References
Anton, H. and Rorres, C. (2005). Elementary Linear Algebra. Applications Version. John Wiley
& Sons.
Cramer, E. and Neslehova, J. (2008). Vorkurs Mathematik. Arbeitsbuch zum Studienbeginn in
Bachelor-Studiengngen. Springer.
Fahrmeir, L., Künstler, R., Pigeot, I. and Tutz, G. (2007). Statistik: Der Weg zur Datenanalyse. 6th ed. Springer, Berlin.
Grimmett, G. and Stirzaker, D. (2001). Probability and Random Processes. 3rd ed. Oxford
University Press, Oxford.
Held, L. (2008). Methoden der statistischen Inferenz. Springer Verlag.
Heuser, H. (2006a). Lehrbuch der Analysis. Teil 1. Ninth ed. Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart.
Heuser, H. (2006b). Lehrbuch der Analysis. Teil 2. Sixth ed. Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart.
Jänich, K. (2008). Lineare Algebra. Mit 110 Testfragen. Springer.
Strang, G. (2005). Introduction to linear algebra. 3rd ed. Wellesley-Cambridge Press, Wellesley,
MA.
Thompson, S. P. and Gardner, M. (1998). Calculus Made Easy. Newly Revised, Updated,
Expanded, and Annotated for its 1998 edition. St. Martin’s Press.
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