Discrete math for computer science students ken bogart dept. This is a text that covers the standard topics in a sophomorelevel course in discrete mathematics. Categorical variable categorical variables contain a finite number of categories or distinct groups. Binomial random variables, repeated trials and the socalled modern portfolio theory pdf 12. Applications and modeling are central to this course of study. Discrete math concerns counting, probability, sophisticated forms of addition, and limit processes over discrete sets.
A visually animated interactive introduction to discrete mathematics. It discusses languages used in mathematical reasoning, basic concepts, and their properties and relationships among them. I emphasize that random variables are more important than their underlying probability spaces. This zybook demonstrates how to translate english descriptions of everyday scenarios into precise mathematical statements that can then be used for formal analysis. Notes on discrete mathematics northwestern university. This is a book about discrete mathematics which also discusses mathematical rea.
Discrete and continuous random variables video khan. It is a very good tool for improving reasoning and problemsolving capabilities. The definition of fy xy x parallels that of pb a, the conditional probability that b will occur, given that a has occurred. Bound and free variables in discrete math mathematics. Here, x is a variable and stands for any object that meets the criteria after the colon. We also introduce common discrete probability distributions. In the preface, feller wrote about his treatment of. Pdf discrete mathematics and its applications seventh. Many mathematical statements say that a property is true for all values of a variable, when values are chosen from some domain. Discrete and continuous random variables video khan academy. The discrete probability density function pdf of a discrete random variable x can be represented in a table, graph, or formula, and provides the probabilities pr x x for all possible values of x. Firstly, to introduce the basic algorithms for computing exactly with integers, polynomials and vector spaces. For example, categorical predictors include gender, material type, and payment method. Schaums outline of theory and problems of discrete.
Cs 70 discrete mathematics and probability theory fall 2009 satish rao,david tse lecture 16 multiple random variables and applications to inference in many probability problems, we have to deal with multiple r. If it can take on a value such that there is a noninfinitesimal gap on each side of it containing no values that the variable can take on. Think of digital watches versus analog watches ones where the second hand loops around continuously without stopping. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. For example, the length of a part or the date and time a payment is received.
Combinatorics, graph theory, the idea of function, recurrence relations, permutations, and set theory are all part of discrete math. Browse other questions tagged discrete mathematics or ask your own question. Discrete variable discrete variables are numeric variables that have a countable number of values between any. But discrete mathematics deals with much more than just integers. Sequences and series are among the most important applications of. If we discretize x by measuring depth to the nearest meter, then possible values are nonnegative integers less. What are categorical, discrete, and continuous variables. Positive examples to prove existential quantification. Discrete mathematics and its applications seventh edition kenneth rosen. Lecture notes on discrete mathematics july 30, 2019. If it can take on a value such that there is a non infinitesimal gap on each side of it. Discrete mathematics and its applications seventh edition by kenneth h rosen. Let m the maximum depth in meters, so that any number in the interval 0, m is a possible value of x.
It looks logical to deduce that therefore, jackson must study discrete math ematics. For these students the current text hopefully is still of interest, but the intent is not to provide a solid mathematical foundation for computer science, unlike the majority of textbooks on the subject. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Variables stand for and can be replaced by elements from their domain. More broadly, it encompasses mathematical structures or processes that consists of individual. We usually refer to discrete variables with capital letters. All other variables in the expression are calledfree variables. Think of digital watches versus analog watches ones where the. If you have a discrete variable and you want to include it in a regression or anova model, you can decide. A binary relation from a to b is a subset of a cartesian product a x b. Although it is usually more convenient to work with random variables that assume numerical values, this. In mathematics, a variable may be continuous or discrete. Richard mayr university of edinburgh, uk discrete mathematics. We will see that the expectation of a random variable is a useful property of the distribution that satis es an important property.
Schaums outline of theory and problems of discrete mathematics pdf. Some of the mathematics that is most relevant to programming is known as discrete mathematics. If it can take on two particular real values such that it can also take on all real values between them even values that are arbitrarily close together, the variable is continuous in that interval. Vesztergombi parts of these lecture notes are based on l. The course extends students application of matrix arithmetic and probability. Variables are seen in almost all math applications beginning with algebra. Math statistics and probability random variables discrete random variables.
The answer is yes, and the pdf is exactly what you say it is. This video looks at the difference between discrete and continuous variables. Discrete mathematics is the foundation for the formal approaches. Lecture notes in discrete mathematics arkansas tech faculty web. The conditional probability density function of y given that x x is if x and y are discrete, replacing pdfs by pmfs in the above is the conditional probability mass function of y when x x. Appeals should be made within one week after graded quiz is returned. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. Discrete random variables alevel statistics revision looking at probability distribution, cumulative distribution and probability density function. In contrast, continuous mathematics deals with objects that vary continuously, e. Discrete probability is quite applied in nature and it seems desirable to expose students to this topic early on. I provide a very elementary account of discrete probability in chapter 5. This book is designed for a one semester course in discrete mathematics for sophomore or.
Expectation, and distributions we discuss random variables and see how they can be used to model common situations. It is quite possible for a sentence containing variables to not be a state ment. Elements of discrete mathematics virginia commonwealth university. Discrete mathematics propositional logic tutorialspoint. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its. The cartesian product a x b is defined by a set of pairs. Constructing a probability distribution for random variable.
Discrete mathematics propositional logic the rules of mathematical logic specify methods of reasoning mathematical statements. This page intentionally left blank university of belgrade. Problems on discrete mathematics1 ltex at january 11, 2007. For the student, my purpose was to present material in a precise, readable manner, with the concepts and techniques of discrete mathematics clearly presented and demonstrated. Discrete mathematics i computer science tripos, part 1a paper 1 natural sciences tripos, part 1a, computer science option politics, psychology and sociology, part 1, introduction to computer science option 200910 peter sewell computer laboratory university of cambridge timestamp. Bound and free variables in discrete math mathematics stack. The zybooks approach less text doesnt mean less learning.
One lowest quiz score will be dropped when calculating net score. The course requires some exposure to the basic concepts of discrete mathematics, but no computing experience. Just like variables, probability distributions can be classified as discrete or continuous. In this lesson, learn why variables are not something to be afraid of or some concept that is impossible to grasp. It deals with variables that can have two discrete values, 0 false and 1 true. Exam questions discrete random variables examsolutions. This is the mathematics of discrete elements, such as symbols, character strings, truth values, and objectsto use a programming term that are collections of properties. Discrete probability distributions if a random variable is a discrete variable, its probability distribution is called a discrete probability distribution. Outline 1 propositions 2 logical equivalences 3 normal forms richard mayr university of edinburgh, uk discrete mathematics.
It is increasingly being applied in the practical fields of mathematics and computer science. We now present three simple examples to illustrate this. Discrete random variables a probability distribution for a discrete r. Discrete mathematics about the tutorial discrete mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Random variables a random variableis a rule that assigns a numerical. X can take an infinite number of values on an interval, the probability that a. Discrete variables a discrete variable is a variable that can only takeon certain numbers on the number line. Discrete mathematics discrete mathematics introduces students to the mathematics of networks, social choice, and decision making. Autumn 2019 discrete mathematics liming pang quiz policy. Problems on discrete mathematics1 chungchih li2 kishan mehrotra3 syracuse university, new york latex at january 11, 2007 part i 1no part of this book can be reproduced without permission from the authors. Spiral workbook for discrete mathematics open textbook.
Let pbe the statement maria learns discrete mathematics. Discrete mathematics and its applications seventh edition. A predicate is a statement that contains variables predicate variables, and. Though there is no time to cover them in this course, discrete mathematics is also concerned with techniques to solve certain types of problems such as how to. Applications and modeling are central to this course of. Examples of objectswith discrete values are integers, graphs, or statements in logic.
Continuous variables are numeric variables that have an infinite number of values between any two values. If you dont want to use measure theory, then you have to take what you say as the definition of the pdf in this setting. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous. Discrete mathematics deals with objects that come in discrete bundles, e.
Greek philosopher, aristotle, was the pioneer of logical reasoning. The earliest method of manipulating symbolic logic was invented by george boole and subsequently came to be known as boolean algebra. There will be a quiz each monday during lecture, except the midterm weeks. Probability is a wonderfully intuitive and applicable. You may contact me if you have any question on quiz grading. A propositional function that does not contain any free variables is a proposition and has a truth value.
Discrete mathematics, second edition in progress upenn cis. Besides reading the book, students are strongly encouraged to do all the. This is the mathematics of discrete elements, such as symbols, character strings, truth values, and objectsto use a programming term that are collections of. Cs 70 discrete mathematics and probability theory multiple. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. Discrete mathematics lecture notes, yale university, spring 1999 l. Spiral workbook for discrete mathematics open textbook library.
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