# Math 203 Mcgill Assignment Of Benefits

**Offered by:**Mathematics and Statistics

**Degree:**Bachelor of Science

### Program Requirement:

This program provides students with a solid training in both computer science and statistics together with the necessary mathematical background. As statistical endeavours involve ever increasing amounts of data, some students may want training in both disciplines.

### Program Prerequisites

Students entering the Joint Major in Statistics and Computer Science are normally expected to have completed the courses below or their equivalents. Otherwise they will be required to make up any deficiencies in these courses over and above the 72 credits of required courses.

- MATH 133Linear Algebra and Geometry3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Systems of linear equations, matrices, inverses, determinants; geometric vectors in three dimensions, dot product, cross product, lines and planes; introduction to vector spaces, linear dependence and independence, bases; quadratic loci in two and three dimensions.

Offered by: Mathematics and Statistics

- 3 hours lecture, 1 hour tutorial
- Prerequisite: a course in functions
- Restriction A: Not open to students who have taken MATH 221 or CEGEP objective 00UQ or equivalent.
- Restriction B: Not open to students who have taken or are taking MATH 123, MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics.
- Restriction C: Not open to students who are taking or have taken MATH 134.

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 140Calculus 13 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Review of functions and graphs. Limits, continuity, derivative. Differentiation of elementary functions. Antidifferentiation. Applications.

Offered by: Mathematics and Statistics

- 3 hours lecture, 1 hour tutorial
- Prerequisite: High School Calculus
- Restriction: Not open to students who have taken MATH 120, MATH 139 or CEGEP objective 00UN or equivalent
- Restriction: Not open to students who have taken or are taking MATH 122 or MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics
- Each Tutorial section is enrolment limited

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 141Calculus 24 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): The definite integral. Techniques of integration. Applications. Introduction to sequences and series.

Offered by: Mathematics and Statistics

- Prerequisites: MATH 139 or MATH 140 or MATH 150.
- Restriction: Not open to students who have taken MATH 121 or CEGEP objective 00UP or equivalent
- Restriction Note B: Not open to students who have taken or are taking MATH 122 or MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics.
- Each Tutorial section is enrolment limited

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

### Required Courses (51 credits)

* Students who have sufficient knowledge in a programming language do not need to take COMP 202 but can replace it with an additional Computer Science complementary course.

** Students take either COMP 350 or MATH 317, but not both.

*** Students take either MATH 223 or MATH 236, but not both.

- COMP 202Foundations of Programming3 Credits*
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Introduction to computer programming in a high level language: variables, expressions, primitive types, methods, conditionals, loops. Introduction to algorithms, data structures (arrays, strings), modular software design, libraries, file input/output, debugging, exception handling. Selected topics.

Offered by: Computer Science

- 3 hours
- Prerequisite: a CEGEP level mathematics course
- Restrictions: COMP 202 and COMP 208 cannot both be taken for credit. COMP 202 is intended as a general introductory course, while COMP 208 is intended for students interested in scientific computation. COMP 202 cannot be taken for credit with or after COMP 250

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- COMP 206Intro to Software Systems3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Comprehensive overview of programming in C, use of system calls and libraries, debugging and testing of code; use of developmental tools like make, version control systems.

Offered by: Computer Science

- Terms
- Instructors
- Joseph P Vybihal
- David P Meger, Gregory L Dudek

- COMP 250Intro to Computer Science3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Mathematical tools (binary numbers, induction, recurrence relations, asymptotic complexity, establishing correctness of programs), Data structures (arrays, stacks, queues, linked lists, trees, binary trees, binary search trees, heaps, hash tables), Recursive and non-recursive algorithms (searching and sorting, tree and graph traversal). Abstract data types, inheritance. Selected topics.

Offered by: Computer Science

- 3 hours
- Prerequisites: Familiarity with a high level programming language and CEGEP level Math.
- Students with limited programming experience should take COMP 202 or equivalent before COMP 250. See COMP 202 Course Description for a list of topics.

- Terms
- Instructors
- Michael Langer
- Martin Robillard

- COMP 251Algorithms and Data Structures3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Introduction to algorithm design and analysis. Graph algorithms, greedy algorithms, data structures, dynamic programming, maximum flows.

Offered by: Computer Science

- 3 hours
- Prerequisite: COMP 250
- Corequisite(s): MATH 235 or MATH 240 or MATH 363.
- COMP 251 uses mathematical proof techniques that are taught in the corequisite course(s). If possible, students should take the corequisite course prior to COMP 251.
- COMP 251 uses basic counting techniques (permutations and combinations) that are covered in MATH 240 and 363, but not in MATH 235. These techniques will be reviewed for the benefit of MATH 235 students.
- Restrictions: Not open to students who have taken or are taking COMP 252.

- Terms
- Instructors
- Jérôme Waldispuhl
- Luc P Devroye

- COMP 273Intro to Computer Systems3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Number representations, combinational and sequential digital circuits, MIPS instructions and architecture datapath and control, caches, virtual memory, interrupts and exceptions, pipelining.

Offered by: Computer Science

- Terms
- Instructors

- COMP 302Programming Lang & Paradigms3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Programming language design issues and programming paradigms. Binding and scoping, parameter passing, lambda abstraction, data abstraction, type checking. Functional and logic programming.

Offered by: Computer Science

- Terms
- Instructors
- Brigitte Pientka
- Prakash Panangaden

- COMP 330Theory of Computation3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Finite automata, regular languages, context-free languages, push-down automata, models of computation, computability theory, undecidability, reduction techniques.

Offered by: Computer Science

- Terms
- Instructors

- COMP 350Numerical Computing3 Credits**
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Computer representation of numbers, IEEE Standard for Floating Point Representation, computer arithmetic and rounding errors. Numerical stability. Matrix computations and software systems. Polynomial interpolation. Least-squares approximation. Iterative methods for solving a nonlinear equation. Discretization methods for integration and differential equations.

Offered by: Computer Science

- Terms
- Instructors

- COMP 360Algorithm Design3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Advanced algorithm design and analysis. Linear programming, complexity and NP-completeness, advanced algorithmic techniques.

Offered by: Computer Science

- Terms
- Instructors

- MATH 222Calculus 33 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 223Linear Algebra3 Credits***
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Review of matrix algebra, determinants and systems of linear equations. Vector spaces, linear operators and their matrix representations, orthogonality. Eigenvalues and eigenvectors, diagonalization of Hermitian matrices. Applications.

Offered by: Mathematics and Statistics

- Fall and Winter
- Prerequisite: MATH 133 or equivalent
- Restriction: Not open to students in Mathematics programs nor to students who have taken or are taking MATH 236, MATH 247 or MATH 251. It is open to students in Faculty Programs

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 235Algebra 13 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Sets, functions and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. Rings, ideals and quotient rings. Fields and construction of fields from polynomial rings. Groups, subgroups and cosets; group actions on sets.

Offered by: Mathematics and Statistics

- Fall
- 3 hours lecture; 1 hour tutorial
- Prerequisite: MATH 133 or equivalent

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 236Algebra 23 Credits***
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Linear equations over a field. Introduction to vector spaces. Linear mappings. Matrix representation of linear mappings. Determinants. Eigenvectors and eigenvalues. Diagonalizable operators. Cayley-Hamilton theorem. Bilinear and quadratic forms. Inner product spaces, orthogonal diagonalization of symmetric matrices. Canonical forms.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 242Analysis 13 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): A rigorous presentation of sequences and of real numbers and basic properties of continuous and differentiable functions on the real line.

Offered by: Mathematics and Statistics

- Fall
- Prerequisite: MATH 141
- Restriction(s): Not open to students who are taking or who have taken MATH 254.

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 314Advanced Calculus3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Derivative as a matrix. Chain rule. Implicit functions. Constrained maxima and minima. Jacobians. Multiple integration. Line and surface integrals. Theorems of Green, Stokes and Gauss. Fourier series with applications.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 317Numerical Analysis3 Credits**
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 323Probability3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Sample space, events, conditional probability, independence of events, Bayes' Theorem. Basic combinatorial probability, random variables, discrete and continuous univariate and multivariate distributions. Independence of random variables. Inequalities, weak law of large numbers, central limit theorem.

Offered by: Mathematics and Statistics

- Prerequisites: MATH 141 or equivalent.
- Restriction: Intended for students in Science, Engineering and related disciplines, who have had differential and integral calculus
- Restriction: Not open to students who have taken or are taking MATH 356

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 324Statistics3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, contingency tables, nonparametric inference, regression, Bayesian inference.

Offered by: Mathematics and Statistics

- Fall and Winter
- Prerequisite: MATH 323 or equivalent
- Restriction: Not open to students who have taken or are taking MATH 357
- You may not be able to receive credit for this course and other statistic courses. Be sure to check the Course Overlap section under Faculty Degree Requirements in the Arts or Science section of the Calendar.

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 423Regression&Anal of Variance3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Least-squares estimators and their properties. Analysis of variance. Linear models with general covariance. Multivariate normal and chi-squared distributions; quadratic forms. General linear hypothesis: F-test and t-test. Prediction and confidence intervals. Transformations and residual plot. Balanced designs.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

### Complementary Courses (21 credits)

12 credits in Mathematics selected from:

* Students take either MATH 340 or MATH 350, but not both.

** MATH 578 and COMP 540 cannot both be taken for program credit.

- MATH 327Matrix Numerical Analysis3 CreditsTaught only in alternate years
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems.

Offered by: Mathematics and Statistics

- Symbols:
- Taught only in alternate years

- Terms
- This course is not scheduled for the
*2018-2019*academic year

- This course is not scheduled for the
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 340Discrete Structures 23 Credits*
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Review of mathematical writing, proof techniques, graph theory and counting. Mathematical logic. Graph connectivity, planar graphs and colouring. Probability and graphs. Introductory group theory, isomorphisms and automorphisms of graphs. Enumeration and listing.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 350Graph Theory and Combinatorics3 Credits*
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Graph models. Graph connectivity, planarity and colouring. Extremal graph theory. Matroids. Enumerative combinatorics and listing.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 352Problem Seminar1 CreditsRequires departmental approval prior to registration
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Seminar in Mathematical Problem Solving. The problems considered will be of the type that occur in the Putnam competition and in other similar mathematical competitions.

Offered by: Mathematics and Statistics

- Prerequisite: Enrolment in a math related program or permission of the instructor. Requires departmental approval.
- Prerequisite: Enrolment in a math related program or permission of the instructor.

- Symbols:
- Requires departmental approval prior to registration

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 410Majors Project3 CreditsRequires departmental approval prior to registration
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): A supervised project.

Offered by: Mathematics and Statistics

- Prerequisite: Students must have 21 completed credits of the required mathematics courses in their program, including all required 200 level mathematics courses.
- Requires departmental approval.

- Symbols:
- Requires departmental approval prior to registration

- Terms
- Instructors
- Djivede A Kelome
- Djivede A Kelome, Gantumur Tsogtgerel

- MATH 427Statistical Quality Control3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Introduction to quality management; variability and productivity. Quality measurement: capability analysis, gauge capability studies. Process control: control charts for variables and attributes. Process improvement: factorial designs, fractional replications, response surface methodology, Taguchi methods. Acceptance sampling: operating characteristic curves; single, multiple and sequential acceptance sampling plans for variables and attributes.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 447Intro. to Stochastic Processes3 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Conditional probability and conditional expectation, generating functions. Branching processes and random walk. Markov chains, transition matrices, classification of states, ergodic theorem, examples. Birth and death processes, queueing theory.

Offered by: Mathematics and Statistics

- Winter
- Prerequisite: MATH 323
- Restriction: Not open to students who have taken or are taking MATH 547.

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 523Generalized Linear Models4 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Modern discrete data analysis. Exponential families, orthogonality, link functions. Inference and model selection using analysis of deviance. Shrinkage (Bayesian, frequentist viewpoints). Smoothing. Residuals. Quasi-likelihood. Contingency tables: logistic regression, log-linear models. Censored data. Applications to current problems in medicine, biological and physical sciences. R software.

Offered by: Mathematics and Statistics

- Winter
- Prerequisite: MATH 423
- Restriction: Not open to students who have taken MATH 426

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 524Nonparametric Statistics4 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Distribution free procedures for 2-sample problem: Wilcoxon rank sum, Siegel-Tukey, Smirnov tests. Shift model: power and estimation. Single sample procedures: Sign, Wilcoxon signed rank tests. Nonparametric ANOVA: Kruskal-Wallis, Friedman tests. Association: Spearman's rank correlation, Kendall's tau. Goodness of fit: Pearson's chi-square, likelihood ratio, Kolmogorov-Smirnov tests. Statistical software packages used.

Offered by: Mathematics and Statistics

- Fall
- Prerequisite: MATH 324 or equivalent
- Restriction: Not open to students who have taken MATH 424

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 525Sampling Theory & Applications4 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Simple random sampling, domains, ratio and regression estimators, superpopulation models, stratified sampling, optimal stratification, cluster sampling, sampling with unequal probabilities, multistage sampling, complex surveys, nonresponse.

Offered by: Mathematics and Statistics

- Prerequisite: MATH 324 or equivalent
- Restriction: Not open to students who have taken MATH 425

- Terms
- This course is not scheduled for the
*2018-2019*academic year

- This course is not scheduled for the
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 545Intro to Time Series Analysis4 Credits
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Stationary processes; estimation and forecasting of ARMA models; non-stationary and seasonal models; state-space models; financial time series models; multivariate time series models; introduction to spectral analysis; long memory models.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- MATH 578Numerical Analysis 14 Credits**
- Fall
- Winter
- Summer

Offered in the:

Mathematics & Statistics (Sci): Development, analysis and effective use of numerical methods to solve problems arising in applications. Topics include direct and iterative methods for the solution of linear equations (including preconditioning), eigenvalue problems, interpolation, approximation, quadrature, solution of nonlinear systems.

Offered by: Mathematics and Statistics

- Terms
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

9 credits in Computer Science selected as follows:

At least 6 credits selected from:

- COMP 424Artificial Intelligence3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Introduction to search methods. Knowledge representation using logic and probability. Planning and decision making under uncertainty. Introduction to machine learning.

Offered by: Computer Science

- Terms
- Instructors

- COMP 462Computational Biology Methods3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Application of computer science techniques to problems arising in biology and medicine, techniques for modeling evolution, aligning molecular sequences, predicting structure of a molecule and other problems from computational biology.

Offered by: Computer Science

- Terms
- Instructors

- COMP 526Probabilistic Reasoning and AI3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Belief networks, Utility theory, Markov Decision Processes and Learning Algorithms.

Offered by: Computer Science

- Terms
- This course is not scheduled for the
*2018-2019*academic year

- This course is not scheduled for the
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- COMP 540Matrix Computations4 Credits**
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Designing and programming reliable numerical algorithms. Stability of algorithms and condition of problems. Reliable and efficient algorithms for solution of equations, linear least squares problems, the singular value decomposition, the eigenproblem and related problems. Perturbation analysis of problems. Algorithms for structured matrices.

Offered by: Computer Science

- Terms
- Instructors

- COMP 547Cryptography & Data Security4 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): This course presents an in-depth study of modern cryptography and data security. The basic information theoretic and computational properties of classical and modern cryptographic systems are presented, followed by a cryptanalytic examination of several important systems. We will study the applications of cryptography to the security of systems.

Offered by: Computer Science

- Terms
- This course is not scheduled for the
*2018-2019*academic year

- This course is not scheduled for the
- Instructors
- There are no professors associated with this course for the
*2018-2019*academic year

- There are no professors associated with this course for the

- COMP 551Applied Machine Learning4 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Selected topics in machine learning and data mining, including clustering, neural networks, support vector machines, decision trees. Methods include feature selection and dimensionality reduction, error estimation and empirical validation, algorithm design and parallelization, and handling of large data sets. Emphasis on good methods and practices for deployment of real systems.

Offered by: Computer Science

- Prerequisite(s): MATH 323 or ECSE 205 or ECSE 305 or equivalent
- Restriction(s): Not open to students who have taken COMP 598 when topic was "Applied Machine Learning"
- Some background in Artificial Intelligence is recommended, e.g. COMP-424 or ECSE-526, but not required.

- Terms
- Instructors

- COMP 564Adv Comput'l Bio Meth&Research3 Credits
- Fall
- Winter
- Summer

Offered in the:

Computer Science (Sci): Fundamental concepts and techniques in computational structural biology, system biology. Techniques include dynamic programming algorithms for RNA structure analysis, molecular dynamics and machine learning techniques for protein structure prediction, and graphical models for gene regulatory and protein-protein interaction networks analysis. Practical sessions with stateof- the-art software.

Offered by: Computer Science

MATH 203: Principles of Statistics 1 Assignment 4 Solutions 1) Problem 4.30 Let X denote the number of points awarded. Then we have that: Outcome of Appeal Number of Cases X Plaintiff trial win - reversed 71 -1 Plaintiff trial win - affirmed/dismissed 240 5 Defendant trial win - reversed 68 -3 Defendant trial win - affirmed/dismissed 299 5 Total 678 – Then, we have that the probability distribution for X is as follows: P (X = −3) = 68 = 0.100 678 P (X = −1) = 71 = 0.105 678 P (X = 5) = 240 + 299 = 0.795 678 The graph for the probability distribution for X is: 1 0.0 0.2 0.4 P(X=x) 0.6 0.8 1.0 Probability Distribution for X −3 −2 −1 0 1 2 3 4 5 x 2) Problem 4.44 From the probability distribution for X we have: X µ = E(X) = xP (X = x) = 1(0.40) + 2(0.54) + 3(0.02) + 4(0.04) = 1.70 x∈Rx 2 E(X ) = X x2 P (X = x) = 12 (0.40) + 22 (0.54) + 32 (0.02) + 42 (0.04) = 3.38 x∈Rx σ 2 = E(X 2 ) − µ2 = 3.38 − 1.702 = 0.49 √ √ σ = σ 2 = 0.49 = 0.7 Thus, the mean is µ = 1.70 and the standard deviation is σ = 0.7. We can interpret the mean as follows: if we take a very large (to infinite) sample of blades of water hyacinth and observe the number of insect eggs on the blades, the sample mean of the number of insect eggs will be 1.70. 3) Problem 4.46 Let X = winnings in the Florida lottery. The probability distribution for X is then: 2 x P (X = x) $-1 22,999,999/23,000,000 $ 6,999,999 1/23,000,000 The expected net winnings is thus: µ = E(X) = (−1)(22, 999, 999/23, 000, 000) + (6, 999, 999)(1/23, 000, 000) = $ − 0.70 So the average winnings of all those who play the lottery is $ - 0.70. That is, if we take a very large (to infinite) sample of individuals playing the Florida lottery and observe their net winnings, the sample mean of that sample will be -0.70. That is, if we independently sample n lottery tickets and gain amounts x1 , . . . , xn , the sample average from the x1 , . . . , xn will converge to µ as n grows larger (to infinity!) As a practical interpretation: on average, if you buy a lottery ticket you will lose money. For interest and an exercise in critical thinking: Note that this interpretation doesn’t quite work in this case. According to Wikipedia, in the Florida Pick-6 Lotto, 6 balls are drawn each Wednesday and Saturday without replacement from 53 numbers. Contestants pick numbers in advance and are required to match all 6 numbers, but 53 the order doesn’t matter. Note that there are = 22957480 different combinations of numbers, 6 confirming the figure given of approximately 1 in 23 million. So, we can consider that our Lotto player perhaps chooses the same set of 6 numbers and plays them at each Lotto drawing. The winning numbers at each drawing are independent (unless the contestant is rigging the lottery somehow!). So, here we can easily consider a sequence of independent realizations x1 , . . . , xn as n goes to infinity: our lotto player simply keeps playing! The tricky part comes in with regard to the grand prize: it changes as time goes on! Once the grand prize is won, it usually resets to a lower number. So, we cannot really consider a sequence of realizations x1 , . . . , xn where n increases to infinity, because once the jackpot is won, the sequence breaks down (they are no longer from the same probability distribution). Additionally, when nobody wins at a drawing, the grand prize amount generally increases. A more technically correct interpretation would be this: If the lotto player were to purchase a ticket with each possible set of 6 numbers out of the 53, the average winnings per ticket would be approximately $−0.70. 4) Problem 4.64 Let X = the number of brands that use tap water. 3 (a) We will check the characteristics of a binomial random variable 1 - This experiment consists of n = 5 identical trials. 2 - There are only 2 possible outcomes for each trial: A brand of bottled water either uses tap water (S) or not (F ). 3 - The probability of S remains the same from trial to trial. In this case, p = P (S) ≈ 0.25 for each trial. 4 - The trials are independent. Since the number of bottles of water from which to sample is large compared to the sample size of 5, the trials are close enough to being independent. 5 - X = the number of brands of bottled water using tap water in 5 trials. Thus, X ≈ Binomial(n = 5, p = 0.25). (b) The formula for finding the binomial probabilities in this case is: 5 P (X = x) = (0.25)x (0.75)5−x for x = 0, 1, 2, 3, 4, 5 x (c) P (X = 2) = 5 2 (0.25)2 (0.75)3 (d) P (X ≤ 1) = P (X = 0) + P (X = 1) = 5 0 (0.25)0 (0.75)5 + 5 1 (0.25)1 (0.75)4 5) Problem 4.136 Let X = the number of beach trees damaged by fungi in 20 trials. Then X is a binomial random variable with n = 20 and p = 0.25. (a) P (X < 10) = 9 X P (X = x) = P (X = 0) + P (X = 1) + ... + P (X = 9) x=0 20 20 20 0 20 1 19 = (0.25) (0.75) + (0.25) (0.75) + ... + (0.25)9 (0.75)19 0 1 9 4 (b) P (X > 15) = 20 X P (X = x) = P (X = 16) + P (X = 17) + P (X = 18) + P (X = 19) + P (X = 20) x=16 20 20 20 16 4 17 3 = (0.25) (0.75) + (0.25) (0.75) + (0.25)18 (0.75)2 16 17 18 20 20 19 1 + (0.25) (0.75) + (0.25)20 (0.75)0 19 20 (c) µ = E(X) = np = (20)(0.25) = 5 6) Problem 5.38 Let X denote the amount of miraculin produced. We are told that X ≈ Normal(µ = 105.3, σ = 8). Then using the standard normal table, we have: (a) 120 − 105.3 ) 8 = P (Z > 1.84) P (X > 120) = P (Z > = P (Z > 0) − P (0 < Z < 1.84) = 0.5 − 0.4671 = 0.0329 (b) 100 − 105.3 110 − 105.3 <Z< ) 8 8 = P (−0.66 < Z < 0.59) P (100 < X < 110) = P ( = P (−0.66 < Z < 0) + P (0 < Z < 0.59) = P (0 < Z < 0.66) + P (0 < Z < 0.59) = 0.2454 + 0.2224 = 0.4678 5 (c) P (X < a) = 0.25 a − 105.3 ⇒ P (Z < ) = P (Z < z0 ) = 0.25 8 ⇒ P (Z < z0 ) = P (Z < 0) − P (z0 < Z < 0) = P (Z > 0) − P (0 < Z < −z0 ) ⇒ P (0 < Z < −z0 ) = P (Z > 0) − P (Z < z0 ) = 0.5 − 0.25 = 0.25 Looking up the area 0.25 in the table, we have that −z0 = 0.67, thus a − 105.3 8 ⇒ a = 8(−0.67) + 105.3 z0 = −0.67 = = 99.94 7) Problem 5.42 Let X denote the carapace length of green sea turtles. We are told that X has a normal distribution with µ = 55.7 and σ = 11.5. Then: (a) The probability of catching a sea turtle that is considered legal is 40 − 55.7 60 − 55.7 <Z< ) 11.5 11.5 = P (−1.37 < Z < 0.37) P (40 < X < 60) = P ( = P (−1.37 < Z < 0) + P (0 < Z < 0.37) = P (0 < Z < 1.37) + P (0 < Z < 0.37) = 0.4147 + 0.1443 = 0.5590 6 Then, the probability of capturing a sea turtle that is considered illegal is P (X < 40) + P (X > 60) = 1 − P (40 < X < 60) = 1 − 0.5590 = 0.4410 (b) We want to find the maximum limit, L, such that only 10% of turtles captured have shell lengths greater than L; thus we want P (X > L) = 0.10. So we have: L − 55.7 ) 11.5 = P (Z > z0 ) P (X > L) = P (Z > = 0.10 We also have that: P (Z > z0 ) = P (Z > 0) − P (0 < Z < z0 ) ⇒ 0.10 = 0.5 − P (0 < Z < z0 ) ⇒ P (0 < Z < z0 ) = 0.50 − 0.1 = 0.40 From the normal table, we can find that z0 ≈ 1.28, so: L − 55.7 11.5 ⇒ L = 1.28 × 11.5 + 55.7 z0 = 1.28 = = 70.42 7 8) Problem 5.90 (a) Let X denote the percentage of body fat in men. We are told that X ≈ Normal(µ = 15, σ = 0.2). Then for any particular man, the probability of being obese is: P (Obese) = P (X ≥ 20) 20 − 15 = P (Z ≥ ) 0.2 = P (Z ≥ 2.5) = P (Z ≥ 0) − P (0 ≤ Z ≤ 2.5) = 0.5 − 0.4938 = 0.0062 Now let Y be the number of men in the U.S. Army who are obese in a sample of 10,000. The random variable Y is Binomial with n = 10000 and p = 0.0062. Then we have that: E(Y ) = µY = np = 10000(0.0062) = 62 p √ σY = npq = 10000(0.0062)(1 − 0.0062) = 7.85 The interval µY ± 3σY is: 10000(0.0062) ± p 10000(0.0062)(1 − 0.0062) = 62 ± 3(7.85) ⇒ (38.45, 85.55) Since the interval lies in the range (0, 10000), we can use the normal approximation. Taking into account the continuity correction, we have: P (Y < 50) = P (Y ≤ 49) 49 + 0.5 − 62 ) ≈ P (Z ≤ 7.85 −12.5 = P (Z ≤ ) 7.85 = P (Z ≤ −1.59) = P (Z ≤ 0) − P (−1.59 ≤ Z ≤ 0) = P (Z > 0) − P (0 < Z < 1.59) = 0.5 − 0.4441 = 0.0559 8 If we do not make the continuity correction, this approximation becomes: P (Y < 50) = P (Y ≤ 49) 49 − 62 ≈ P (Z ≤ ) 7.85 = P (Z ≤ −1.66) = P (Z ≤ 0) − P (−1.66 ≤ Z ≤ 0) = P (Z > 0) − P (0 ≤ Z ≤ 1.66) = 0.5 − 0.4515 = 0.0485 Note that to validate the use of the normal approximation, we could also check that n > 30, np = 62 > 5 and nq = 9938 > 5. (b) In part a) we found that the probability of being obese was P (X > 20) = 0.0062, so that 0.62% of the general population of American men are obese. In the sample of 10,000 army men, there are 30 obese men so that pˆ = 0.003, i.e. 0.3% of men in the American army are obese (since the sample size is quite large, pˆ is a good approximation for the true percentage in the American army men population). Since 0.003 < 0.0062, we can indeed conclude that the army was successful in reducing the percentage of obese men below the percentage in the general American men population. 9) Problem 6.47 Let Xi denote the bacterial count of the ith health care worker who washes their hands. We are given the mean (µ = 69) and standard deviation (σ = 106). Then the probability that the sum of the bacterial counts from 50 hand washers is greater than 1510 is equivalent to: P 50 X i=1 ! Xi > 1510 P50 =P i=1 Xi 50 9 1510 > 50 ! ¯ > 30.2). = P (X ¯ is approximately normally As the sample size is large (n > 30), by the Central Limit Theorem, X √ distributed with mean µ = 69 and standard deviation σ/ 50 = 14.99 ¯ X −µ 30.2 − 69 √ > 14.99 σ/ n = P (Z > −2.59) ¯ > 30.2) = P P (X = P (−2.59 < Z < 0) + P (Z > 0) = P (0 < Z < 2.59) + P (Z > 0) = 0.4952 + 0.5 = 0.9952 The approximate probability that the total bacterial count for the sample of 50 hand washers is greater than 1510 is 99.5%. Alternatively, if we consider the bacterial count per hand, we obtain the following: Let Xi denote the bacterial count on one hand of the ith health care worker who washes their hands. We are given the mean (µ = 69) and standard deviation (σ = 106) of the bacterial count per hand. In the random sample, there are 50 health care workers, so a total of 100 hands. Then the probability that the sum of the bacterial counts from the 50 hand washers is greater than 1510 is equivalent to: P 100 X ! Xi > 1510 P100 =P i=1 i=1 Xi 100 1510 > 100 ! ¯ > 15.1). = P (X ¯ is approximately normally As the sample size is large (n > 30), by the Central Limit Theorem, X √ distributed with mean µ = 69 and standard deviation σ/ 100 = 10.6, so we have: ¯ 15.1 − 69 X −µ √ > 10.6 σ/ n = P (Z > −5.08) ¯ > 15.1) = P P (X = P (−5.08 < Z < 0) + P (Z > 0) = P (0 < Z < 5.08) + P (Z > 0) ≈ 0.5 + 0.5 = 1 10

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