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Introduction to LATEX Part II: Writing a Technical Paper This Frame Intentionally Left Blank Monson H. Hayes [email protected] Chung-Ang University Seoul, Korea This material is the property of the author and is for the sole and exclusive use of his students. It is not for publication, nor is it to be sold, reproduced, or generally distributed. M.Hayes (CAU-GT) November 08, 2011 1 / 54 M.Hayes (CAU-GT) Getting Started with your LATEX Technical Paper Paper Preamble First Step: Set up the preamble, which is the area between \documentclass and \begin{document}. For the sample paper, the preamble is quite simple: The preamble includes the definition of the document class with options, November 08, 2011 2 / 54 %\documentclass[journal,onecolumn,twoside]{IEEEtran} \documentclass[10pt,twocolumn,twoside]{IEEEtran} \usepackage[pdftex]{graphicx} \usepackage{amsmath,amssymb} \usepackage{setspace} \usepackage{subfigure} \def\pr{{\rm P}} \documentclass[journal,onecolumn]{IEEEtran} Global style commands. \setlength{\parindent}{0pt} Packages that you want to include, \usepackage[pdftex]{graphicx} \begin{document} Your own special features and definitions \def\pr{{\rm P}} M.Hayes (CAU-GT) November 08, 2011 3 / 54 M.Hayes (CAU-GT) November 08, 2011 4 / 54 LATEX Paper Header NS ON LATEX 1 For the sample paper, the header is straightforward: Random Variables: An Overview Monson Hayes, Fellow, IEEE Chung-Ang University Seoul, Korea [email protected] A \title{Random Variables: An Overview} \author{Monson Hayes,~\IEEEmembership{Fellow,~IEEE} \\ Chung-Ang University \\ Seoul, Korea \\ \textit{[email protected]}} \markboth{IEEE Transactions on \LaTeX\ } {Hayes} IEEE TRANSACTIONS ON LTEX (Invited Paper) 1 Random Variables: An Overview paper introduces the concept of a random s nothing more than a variable whose numeric ed by the outcome of an experiment. To describe that are associated with these numeric values d conceptually useful manner, the probability M.Hayes (CAU-GT) probability density function are introduced. In y important concepts of conditional distribution ensity functions are presented. Then, the expeced values of random variables are introduced, e framework describing random variables in PaperforAbstract . Throughout the chapter, a variety of different s that are important in applications will be \IEEEspecialpapernotice{(Invited Paper)} \maketitle which a geometric probability law, and the random variable N is referred to as a geometric random variable. It is straightforward to show that this probability assignment satisfies the Monson Hayes, Fellow, IEEE three probability axioms. Pn 5 / 54 π 2 08, 2011 = Chung-Ang style M.Hayes (CAU-GT) limn→∞ k=1 k12 November University 6 . And this is display November 08, 2011 n X π 2 Korea 1 Seoul, = (2) 2 n→∞ k 6 [email protected] k=1 LATEX Given the probability assignment for the random variable N , it is then easy to find the(Invited probability Paper) of any event that is defined in terms of values of X. For example, the \begin{abstract} probability that the number of α particles is less than some number, N0 , This paper introduces the concept of a random variable, may be found as follows. Since the event {N < N0 } is the I. I NTRODUCTION λ > 0. Abstract—This paper introduces the concept of a random for some union of the events {N = k} for k = 0, 1, . . . , N0which − 1, is nothing more than a variable whose numeric variable, which is nothing more than a variable whose numeric value determined by the outcome experiment. Given this is probability assignment for N , of it isanthen easy to in the development of probability theory is to N[ 0 −1 value is determined by the outcome of an experiment. To describe To describe the probabilities that are associated with ncept of a random variable. Although perhaps {N = find n} the probability of any event that is defined in terms of {N < N0 } = the probabilities that are associated with these numeric values t like something difficult, random variables these numeric values in a concise and conceptually n=0 values of N . For example, the probability that the number of useful in a concise and conceptually quite simple. Given a sample space Ω cor- useful manner, the probability the probability distribution and probability and since these events are mutually exclusive, thenmanner, it follows α particles is less than some number, N0 , may be found as distribution andthisprobability ome random experiment, sample spacedensity function are introduced. density function are introduced. from Axiom 3 that ary events, ω ∈ the Ω, and when an experiment Then, moment generatingis function is defined, and several follows. Sincethe themoment event {N < N0 } isfunction the union the events Then, generating is of defined, and cific elementary event outcome) examples are(experimental given. Finally, the concept of0 }a =correlation function P{N < N P{N = 0}+P{N = 1}+·{N · ·+P{N = N −1} 0 = several k} for k examples = 0, 1, . . are . , Ngiven. 0 − 1, en theseand experimental outcomes are numeric correlation matrices is introduced. Finally, the concept ofN a−1correlation function and the temperature of a fluid, the velocity of Therefore, 0 correlation matrices is [ introduced. NX NX 0 −1 0 −1 e value of a commodity or stock, we have {N = n} {N < N } = 0 n \end{abstract} P{N < N0 } = P{N = n} = 0.9(0.1) le. In some experiments, the outcomes may I. I NTRODUCTION n=0 n=0 n=0 %\doublespace such as the gender or the blood type of a at randomThe from concept within a given This last sum be evaluated geometric of a population, random variable is amay simple one, using and theand sinceseries, these events are mutually exclusive, then of the flip of a coin. In these cases, a random N −1 oneM.Hayes that(CAU-GT) is important. Although perhaps sounding at 1first X November 08, 2011 / 54 M.Hayes (CAU-GT) N0 −1 November 08, 2011 NX − α7 N 0 −1 formed simply by mapping these non-numeric X λn αn = 6 / 54 lim like something difficult, random variables are conceptually −λ 8 / 54 in a concise and conceptually useful manner, To thedescribe probability value is determined by the outcome of an experiment. follows. α Since the eventis{Nless < Nthan unionnumber, of the events 0 } is the particles some N0 , may be found as distribution and probability densitywithfunction are values introduced. the probabilities that are associated these numeric {N = k} for k = 0, 1, . . . , N − 1, 0 in concise andgenerating conceptually function useful manner, the probability Then, Introduction thea moment is defined, and several follows. {N < N0 } is the union of the events Paper LATESince X the event distribution and Finally, probability function are introduced. 0 −1 examples are given. thedensity concept of a correlation function {N = k} for k =N0, [ 1, . . . , N0 − 1, Then, the moment generating function is defined, and several {N < N0 } = {N = n} and correlation matrices is introduced. examples are given. Finally, the concept of a correlation function and correlation matrices is introduced. I. I.I NTRODUCTION I NTRODUCTION n=0 N[ 0 −1 and since these events are mutually {N <exclusive, N0 } = then {N = n} NX NX \section{Introduction} 0 −1 n=0 0 −1 λn −λ e P{N < NThe P{Nof= an}random = variable is a simple one, and 0 } =concept n! The concept of of a random a simple one, and and since these The concept a random variable variable is is a simple one, and events are mutually exclusive, then n=0 n=0 is important. one that one is thatimportant. is important.Although Although perhaps sounding at firstat first one that perhaps sounding This last sum may be evaluated usingNX the Although perhaps sounding at first Nlike difficult, 0 −1 something 0 −1following n X like something difficult, random variables are conceptually λsimple. −λ like something difficult, random variables are conceptually random variables are conceptually quite k e P{N = + 1, λ)P{N = n} = quite simple. Given a sample space Ω corresponding to some X λn<−λN0 }Γ(k n! Given a sample space $\Omega $ corresponding to some random quite simple. Given a sample space Ω corresponding to some e = random experiment, this sample space contains elementary n=0 n=0 n! k! n=0 experiment, this sample space contains elementary events, ω ∈ Ω, and an experiment is performed, a randomevents, experiment, thiswhen sample space contains elementary This last sum may be evaluated using the following $\omega \in (experimental outcome)is is observed. where Z ∞\Omega $, and when an experiment is performed, events,specific ω ∈elementary Ω, and event when an experiment performed, a aΓ(k, specific When these experimental outcomes are numeric values, such kxk−1 en−x dx event (experimental outcome) is λ) = elementary X specificas elementary event (experimental outcome) is observed. λ −λ Γ(k + 1, λ) λ the temperature of a fluid, the velocity of a particle, or the observed. e = When value theseofexperimental outcomes are numeric values, such a commodity or stock, we have a random variable. In n! k! n=0 experiments, outcomes be non-numeric, such as or the as the some temperature of the a fluid, themay velocity of a particle, III. P ROBABILITY M ASS F UNCTION theagender or the blood type ofwe a person at random value of commodity or stock, have selected a random variable. In where Z ∞ from within a given population, or the outcome of the flip of a Discrete random variables have a discrete set ofk−1 possible some experiments, the outcomes may be non-numeric, such as x e−x Γ(k, λ)of=these outcomes coin. In these cases, a random variable may be formed simply M.Hayes (CAU-GT) November 08, 2011 outcomes, 9 / 54 M.Hayes (CAU-GT) November 08, 2011 10 / 54 xk , and the probabilities aredx λ the gender or the blood type of a person selected at random by mapping these non-numeric outcomes to real numbers We denoted by pk , begin this paper population, by introducing of of a random from within a given or an theexample outcome the flip of a P{X = xk } = pk variable, and how a probability assignment may be made on coin. In these cases, a random variable may be formed simply III. referred P ROBABILITY M ASS F UNCTION These probabilities to as probability masses Section II experimental with Numbered LATEare X often the outcomes.Equation and Footnote by mapping these non-numeric outcomes to real numbers since We they represent discrete (point) masses of probability at Discrete variables havecharges a discrete set of possible specific alongrandom the real axis, just as point in begin this paper II.byP ROBABILITY introducing an example of a random locations A SSIGNMENTS \section{Probability Assignments} electrostatics are used to represent discrete charges distributed x , and the probabilities of these outcomes are variable,Let and how a probability assignment be made on outcomes, Let $N$kbe a variable that represents the number N be a variable that represents the number may of α particles along a denoted line. These probabilities, written or plotted as a pk , $ particles of by $\alpha that are counted over a given period of that are counted over a given period of time. The ensemble the experimental outcomes. function of x, is called the probability mass function (PMF). time. for N is the set of non-negative integers P{Xfunctions = xk }mathemat= pk In order to express probability The ensemble formass $N$2 is the set of non-negative integers which is defined as \} \] ically, we introduce the delta function, E = {0, 1, 2, . . .} N \[ {\cal E}_N \{ 0,referred 1, 2, \ldots II. P ROBABILITY A SSIGNMENTS These probabilities are =often to as probability masses follows: Since the number of outcomes is unknown until we actually make a Since number ofthat outcomes is unknown until we of actually 1 ; discrete n=0 they represent (point) masses of probability at Let N bethe a variable represents the number α particles since count, δ[n] =then $N$ is a random variable. make a count, then N is a random variable. In many cases, 0 ; n = 6 0 along real axis, to justmodel as point in In locations many cases, it isthe appropriate $N$ ascharges a that are counted over a given period of time. The ensemble specific it is appropriate to model N as a Poisson random variable \emph{Poisson random variable} where\footnote{Note that with this Shifted delta functions may be used used totorepresent functions that charges distributed electrostatics are represent discrete for N where is the1 set of non-negative integers n probability assignment it is assumed that the number of particles have a value of one at other values of n. For example, δ[n−1] λ −λ a line. These probabilities, written or plotted as a P{N = n} = e n≥0 (1) is equal along may be arbitrarily large and, in fact, approach infinity.} to one when n = 1 and equal to zero for all other EN = {0,n!1, 2, . . .} of x, the probability mass function (PMF). \begin{equation} values offunction n. Therefore, for is an called integer-valued discrete random for some λ > 0. \pr \{ N = n \} variable X In withorder to express probability mass functions mathematSince theGiven number of outcomes is unknown we to actually this probability assignment for N , it isuntil then easy = \frac {\lambda ^n}{n!} e^{-\lambda } \qquad n\geq 0 introduce the < delta function,2 which is defined as the probability event that variable. is defined in make afindcount, then N ofisany a random In terms manyof cases, ically, P{X we = n} = pX [n] ; −∞ n<∞ \label{eq:prob_assgn} it is appropriate as a Poisson variable follows: \end{equation} 1 Note that withto thismodel probabilityN assignment it is assumed random that the number ; n=0 2 In digital signal processing, for someδ[n] $\lambda > 0$. 1 in fact, approach infinity. is referred where1of particles may be arbitrarily large and, δ[n] to=as the unit sample function. n 0 ; n 6= 0 λ −λ M.Hayes (CAU-GT) M.Hayes (CAU-GT) November 08, 2011 11 / 54 P{N = n} = e n≥0 (1) November 08, 2011 12 / 54 @gatech.edu ted Union Paper) LATEX bles: An Overview m values of N . For example, the probability that the number of yes, αFellow, IEEE ric particles is less than some number, N0 , may be found as be University Ang follows. Since the event {N < N0 } is the union of the events es oul, Korea ty {N = k} for k = 0, 1, . . . , N0 − 1, @gatech.edu d. N0 −1 al ited on m ric be nd es rst ty ly ed. me ral ry on Paper) {N < N0 } = [ n=0 {N = n} and since these events are mutually exclusive, then values of N . For example, that number of NX 0 −1 the 0 −1 the probabilityN X λn −λ α particles number, N0 , may bee found as P{N = n} = P{N is < less N0 } than = some n!of the events follows. Since the eventn=0 {N < N0 } is the union n=0 {N =last k}sum for kmay = 0,be1,evaluated . . . , N0 −using 1, the following This M.Hayes (CAU-GT) N[ 0 −1 k n X λ Γ(k + 1,=λ)n} −λ {N < Ne0 } == {N n! n=0 k! n=0 November 08, 2011 13 / 54 a and events a Sumare mutually andIntegral since these exclusive, then d. An where Z ∞ ch NX NX −1 0 −1 0= Γ(k, λ) xk−1 e−x dx λn −λ λ he e P{N = n} = P{N < N0 } = n! nd In n=0 n=0 rst as This last sum be evaluatedMusing following III.may P ROBABILITY ASS Fthe UNCTION ly m k mea X λn −λ have Γ(ka +discrete 1, λ) set of possible Discrete random variables e = ry ly outcomes, xk , and then! probabilities k!of these outcomes are n=0 Wea denoted by pk , d. where Z x∞k } = pk m P{X = ch Γ(k, λ) = xk−1 e−x dx on to as probability masses λ he These probabilities are often referred In since they represent discrete (point) masses of probability at along the real axis, just as point charges in as specific locations III. P ROBABILITY M ASS F UNCTION m electrostatics are used to represent discrete charges distributed es a line. Thesevariables probabilities, as a f a along Discrete random have awritten discreteorsetplotted of possible le M.Hayes (CAU-GT) November 08, 2011 15 / 54 of xx,k , isand called probability of mass function (PMF). ly function outcomes, the the probabilities these outcomes are For example, the probability that the number of $\alpha $ particles is less than some number, $N_0$, may be found as follows. Since the event $\{ N < N_0 \}$ is the union of the events $\{ N=k \}$ for $k = 0, 1, \ldots , N_0-1$, \[ \{ N < N_0 \} = \bigcup _{n=0} ^{N_0-1} \{N = n\} \] and since these events are mutually exclusive, then \begin{equation*} \pr \{N < N_0 \} = \sum _{n=0} ^{N_0-1} \pr \{ N=n \} = \sum _{n=0} ^{N_0-1} \frac {\lambda ^n}{n!} e^{-\lambda} \end{equation*} M.Hayes (CAU-GT) November 08, 2011 14 / 54 LATEX This last sum may be evaluated using the following \[ \sum _{n=0} ^{k} \frac {\lambda ^n}{n!} e^{-\lambda } = \frac {\Gamma(k+1,\lambda)}{k!} \] where \[ \Gamma (k,\lambda ) = \int _{\lambda }^{\infty} x^{k-1}e^{-x}dx \] M.Hayes (CAU-GT) November 08, 2011 16 / 54 electrostatics are used to represent discrete charges distributed es along a line. These probabilities, written or plotted as a le Arrays with Invisible Delimiters function of x, is called the probability mass function (PMF). In order to express probability mass functions mathematically, we introduce the delta function,2 which is defined as 2 follows: ly 1 ; n=0 δ[n] = es, 0 ; n 6= 0 le the PMF may be written as Shifted delta functions may be used to represent functions that ∞ have a value of one at other X values of n. For example, δ[n−1] p [k]δ[n − k] 1) is equal to one pwhen X (n) = n = 1 and Xequal to zero for all other k=−∞ values of n. Therefore, for an integer-valued discrete random For example, a Bernoulli Random Variable X has an ensemble to variable X with of possible outcomes of zero and one of P{X = n} = pX [n] ; −∞ < n < ∞ EX = {0, 1} ber 2 In digital signal processing, δ[n] is referred to as the unit sample function. where M.Hayes (CAU-GT) P{X = 0} = p ; P{X = 1} = 1 −November p 08, 2011 17 / 54 Therefore, the probability mass function for X may be expressed ininterms of delta functions as follows: Fractions Equations pX (n) = pδ[n] + (1 − p)δ[n − 1] Another example is the The Geometric Random Variable that has an ensemble equal to the set of all positive integers EX = {1, 2, 3, . . .} with a probability law given by P{N = k} = ( 12 )k ; k > 0 The probability mass function for this random variable is pN (n) = ∞ X ( 12 )k δ[n − k] k=1 Another random variable that occurs frequently in applications is (CAU-GT) one that corresponds to the number ofNovember successes, N M.Hayes 08, 2011 19 /, 54 in n Bernoulli trials, with the probability of a success being LATEX In order to express probability mass functions mathematically, IEEE TRANSACTIONS ON LATEX we introduce the \emph{delta function},\footnote{In digital signal processing, $\delta [n]$ is referred to as the unit sample function.} which is defined as follows: \[ \delta [n] Density = \left\{Function \begin{array}{cl} Name 1 & \ ; \ n = 0 \\ 0 & \ ; \ n \neq 0 \end{array} \right. \] Shifted delta functions be −λx used to represent functions Exponential fX (x)may = λe that have a value of one at 1 other−α|x−m| values of $n$. For example, Laplace fX (x)to=one 2 αe $\delta [n-1]$ is equal when 2$n=1$ and equal to zero for 2 allRayleigh other values of fX$n$. (x) = α2 xe−α x /2 , x ≥ 0 Therefore, discrete random variable $X$ Uniformfor an finteger-valued X (x) = 1/(b − a), b ≤ x ≤ a with \[ \pr \{ X = n \} = p_X[n] \ ; \ -\infty < n < \infty \] TABLE I A TABLE OF COMMON AND IMPORTANT RANDOM VARIABLES . M.Hayes (CAU-GT) November 08, 2011 18 / 54 LATEX the distribution function is the integral of the Conversely, density function, Z x FX (x) = fX (α)dα (2) −∞ Another example is the \emph{The Geometric Random Variable} Two properties of the density are:of all positive that has an ensemble equalfunction to the set 1) integers fX (x) ≥ 0 for all x. Z ∞\[ {\cal E}_X = \{ 1, 2, 3, \ldots \} \] with a probability law given by 2) fX (x)dx = 1 \[ −∞ \pr \{ N = k \} = (\tfrac{1}{2})^k \ ; \ k > 0 \] probability mass function for monotonicity this random variable TheThefirst property follows from the of the is \[ p_N(n) = \sum _{k=1} ^{\infty } distribution function. Specifically, since FX (x) is a non(\tfrac {1}{2})^k \delta [n - k] \] decreasing function of x, then the derivative will always be non-negative for all x. The second property follows by letting x → ∞, in Eq. (2, Z ∞ M.Hayes (CAU-GT) fX (α)dα = FX (∞) = 1 November 08, 2011 −∞ 20 / 54 X ∞ X Another example is the The Geometric Random Variable that pNPhantoms (n) = ( 12 )k δ[n − k] Combinatorics and has an ensemble equal to the set of all positive integers k=1 distribution function. Specifically, since FX (x) is a nonZ x decreasing functionFX of(x) x, = then thefXderivative (α)dα will always be (2) AT X L E non-negative for all x. The second −∞ property follows by letting x→ ∞,properties in Eq. (2,of the density function are: Two Z ∞ variable that occurs frequently in Another random 1) fapplications all x. that corresponds to the number of X (x) ≥ 0 foris fXone (α)dα = FX (∞) = 1 Z ∞ EX = {1, 3, . . .}frequently in applicaAnother random variable that2,occurs tions one that corresponds to the number of successes, N , with ais probability law given by successes,−∞ $N$, in $n$ Bernoulli trials, with the in n Bernoulli trials, with the probability of a success being 2) probability fX (x)dx 1 k of=a 1success being equal to $p$. ) ; k > Distribution 0 = (a2Binomial equal to p. In thisP{N case,=Nk}has with It isIn−∞ important to understand that fX (x) is Distribution} a probability this case, $N$ has a \emph{Binomial The first property follows from the monotonicity of by the densitywith and not a probability. Probabilities are found n for The probability mass function k this random n−k variable is pN (k) = P{N = k} = p (1 − p) ; 0≤k≤n distribution\[ FX (x)of isthea event nonintegrating fXfunction. (x). For example, probability p_N(k) = Specifically, \pr \{ N =the k since \} ∞ k X decreasing of x, then the derivative always = \dbinom{n}{k} p^k(1-p)^{n-k} \ ; \be {X ≤ x} is function given by the integral in Eq. (2), and will the probability pN (n) = ( 12 )k δ[n − k] n 0 ≤ \leq n \] by non-negative second follows by letting where k is the number of combinations of n objects that are of the event for {x1all≤x.XThe x2k} \leq isproperty found evaluating the k=1 where $\tbinom{n}{k}$ is the number of combinations of x → ∞, in Eq. (2, taken k at a time, and is defined by integral Z x2 at a time, and is defined $n$ objects Z ∞that are taken $k$ that occurs frequently in applicaAnother random variable n n! P{x ≤ X ≤ x } = fX (α)dα (3) by 1 2 tions is one that corresponds f (α)dα = F (∞) = 1 = to the number of successes, N , X X x \[ \dbinom{n}{k} = \frac1 {n!}{k!(n-k)!} \] − k)! of a success being k thek!(n −∞ in n Bernoulli trials, with probability Alternative notations include Unlike the distribution function, which$C(n,k)$, is constrained to have n equal to p. notations In this case, N has a Binomial Alternative include C(n, k), n Ck , Distribution Ck , and Cknwith . It is${}_nC_k$, important ${}^nC_k$, to understand fX (x) is a probability and that $C^n_k$. values between zero and one, there is no such constraint on density and not a probability. Probabilities are found by Interesting problems that are sometimes challenging to n k function. large08, 2011 22 / 54 n−k November 08, 2011 M.Hayes (CAU-GT) 21 / 54 the densityM.Hayes (CAU-GT) In fact, fX (x) may be arbitrarily pN (k) P{Nsuch = k} ; 0≤k≤n integrating fX (x). For example, the probability of theNovember event solve, are = those as = k p (1 − p) as long as the integral of fX (x) over all x is equal to one (the X {X ≤ x} is given by the integral in Eq. (2), and the probability area under the curve is unity). is odd}of=combinations P{Nof=nn} where nk is P{N the number objects that are of the event {x1 ≤ X ≤ x2 } is found by evaluating the A table 0≤n≤∞ Substack LATEXof common random variables is given in Table ??. taken k atCommand a time, and is defined by integral Z x2 nodd n n! P{x1 ≤ X ≤ x2 } = fX (α)dα (3) V. M OMENT G ENERATING F UNCTIONS = x1 k!(n − k)! k IV. P ROBABILITY D ENSITY F UNCTION The complete statistical characterization of a random Unlike the distribution function, which is constrained to varihave Alternative notations include C(n, k), n Ck , n Ck , and Ckn . able of between the continuous type of its values zero and one,requires there isthe no specification such constraint on Interesting problems that variable, are sometimes challenging to probability Interesting problems that are sometimes challenging For a continuous random the counterpart of the density function. way to the density distribution function. Inorfact, fX (x) may beAnother arbitrarily large to solve, are those such as solve, are those as is the probability density function. provide probability mass such function thisthestatistical characterization the characteras long as integral of fX (x)isover allisxwith is equal to one (the \[ \pr \{ N \text{ odd}\} jωX The probability density function,X fX (x), of a continuous ran- istic is the expected value of e area function, under thewhich unity). P{N = n} P{N is odd} = =curve \sum is_{\substack{0 \leq n \leq \infty \\[0.5ex] Z dom variable X is the derivative of the probability distribution A table of common random variables is given in Table ??. n∞\text{ odd}}} 0≤n≤∞ jωX function, MX (jω) =\pr E{e = \] ejωx fX (x)dx n odd \{N = }n\} dFX (x) −∞ fX (x) = V. M OMENT G ENERATING UNCTIONS dx where ω √ is a variable taking on any realFnumber between ±∞ jω IV. P ROBABILITY D ENSITY F UNCTION where −1. Since e =characterization cos ω + j sin ω, of thea characteristic andThe j =complete statistical random variFX (x) = P{X ≤ x} function is, in general, type a complex-valued function of ω. It able of the continuous requires the specification of its For a continuous random variable, the counterpart of the probability distribution or density function. Another way to probability mass function is the probability density function. provide this statistical characterization is with the characterM.Hayes (CAU-GT) November 08, 2011 23 / 54 M.Hayes (CAU-GT) November 08, 2011 24 / 54 The probability density function, fX (x), of a continuous ran- istic function, which is the expected value of ejωX le ble LATEX Tables IEEE TRANSACTIONS ON LATEX Name Density Function IEEE TRANSACTIONS ON LATEX −λx Exponential fX (x) = λe 1 −α|x−m| Laplace fDensity Name X (x) = Function 2 αe 2 2 Rayleigh fX (x) = α2 xe−α x /2 , x ≥ 0 Uniform − a), b ≤ x ≤ a Exponential ffXX(x) (x)==1/(b λe−λx 1 Laplace fX (x) = 2 αe−α|x−m| 2 2 Rayleigh fX (x) = α2 xe−α x /2 , x ≥ 0 TABLE I Uniform = 1/(b − a), b ≤ x≤a . X (x) A TABLE OF COMMONfAND IMPORTANT RANDOM VARIABLES xA M.Hayes TABLE (CAU-GT) OF TABLE I November 08, 2011 COMMON AND IMPORTANT RANDOM VARIABLES . 25 / 54 Conversely, the distribution function is the integral of the density function, Z x exSeparation at Enumeration withFItem fX (α)dα X (x) = Conversely, the distribution function is the integral of (2) the hat \begin{table}[t] \begin{center} \begin{tabular}[t]{|ll|} \hline\hline Name & Density Function \\ \hline\hline & \\ Exponential & $f_X(x) = \lambda e^{-\lambda x}$ \\ Laplace & $f_X(x) = \frac{1}{2}\alpha e^{-\alpha |x - m|}$ \\ Rayleigh & $f_X(x) = \alpha ^2 x e^{-\alpha ^2x^2/2}, \ x \geq 0$ \\ Uniform & $f_X(x) = 1/(b-a), \ b \leq x \leq a$ \\ & \\ \hline \end{tabular} \end{center} \caption{A table of common and important random variables.} \label{table:RandomVariables} \end{table} M.Hayes (CAU-GT) November 08, 2011 26 / 54 LATEX −∞ density function, Two properties of the density Z xfunction are: 1) fX (x) ≥ 0 forFXall(x) x.= fX (α)dα Z ∞ −∞ 2)Two properties fX (x)dx 1 density function are: of=the (2) −∞ 1) ffirst ≥ 0 for follows all x. from the monotonicity of the The X (x)property Z ∞ distribution function. Specifically, since FX (x) is a non2) fX (x)dxof=x, 1 then the derivative will always be decreasing function −∞ non-negative all x. The second follows by letting The first for property follows fromproperty the monotonicity of the xdistribution → ∞, in Eq. (2, function. Specifically, since FX (x) is a nonZ ∞ decreasing function of x, then the derivative will always be f (α)dα = FXproperty (∞) = 1follows by letting non-negative for all x.XThe second aN, −∞ ng x → ∞, in Eq. (2, hcaIt is importantZ to ∞ understand that fX (x) is a probability M.Hayes and (CAU-GT)not a probability. Probabilities are 08, 2011 by 27 / 54 N , density fX (α)dα = FX (∞) = 1 Novemberfound Two properties of the density function are: \begin{enumerate}\setlength{\itemsep}{4pt} \item $f_X(x) \geq 0$ for all $x$. \item ${\displaystyle \int _{-\infty } ^{\infty } f_X(x) dx } = 1$ \end{enumerate} M.Hayes (CAU-GT) November 08, 2011 28 / 54 may be recognized that, except for the sign in the exponent, jω is the Fourier thesign probability density Mmay X (ebe)recognized that,transform except forofthe in the exponent, Left and Right Brackets function. useful in the analysis MX (ejωJust ) is as theFourier Fouriertransforms transform are of the probability density offunction. signals and linear systems, the characteristic function is also Just as Fourier transforms are useful in the analysis a of useful tool inlinear probability andtherandom variables. Sinceis also signals and systems, characteristic function Z ∞ Z ∞ a useful tool in probability and random variables. Since |MX (jω)| = Z ∞ejωx fX (x)dx ≤ Z ∞ ejωx fX (x) dx −∞ −∞ jωx e fX (x) dx |MX (jω)| = Z ∞ ejωx fX (x)dx ≤ Z ∞ ejωx |fX (x)| dx =−∞ = Z −∞ Z ∞ fX (x)dx = 1 ∞ −∞ ejωx |fX (x)| dx = −∞ fX (x)dx = 1 = then the characteristic and will always −∞function is well-defined −∞ exist probabilityfunction densityisfunction. thenfor the any characteristic well-defined and will always exist for any probability density function. A. Exponential Random Variable A.If Exponential Random random Variablevariable with parameter λ, X is an exponential If X is an exponential variable f (x) random = λe−λx u(x) with parameter λ, ∂ FX,Y (x, y) 2 ∂x∂y ∂ LAT X fX,Y (x, y) = FX,Y (x, y) where E ∂x∂y FX,Y (x, y) = P{X ≤ x, Y ≤ y} where function, fX,Y (x, y) = (4) (4) FX,Y (x, y) = P{X ≤ x, Y the ≤ joint y} distribution provided the derivatives exist. Conversely, \begin{array*} function thederivatives integral of the Conversely, joint densitythe function, providedisthe joint distribution |M_X(j\omegaexist. )| Z x Z y function is the&= integral the joint density} function, \left|of\int _{-\infty ^{\infty } FX,Y (x, y) = Z x Z ye^{j\omega fX,Y (α, β)dαdβ x} f_X(x) dx (5) \right| −∞ −∞ } β)dαdβ ^{\infty } \left| FX,Y (x,\leq y) = \int _{-\infty fX,Y (α, (5) Given the joint density or distribution function,x}the probability e^{j\omega f_X(x) \right| dx \\ −∞ −∞ &= \int _{-\infty } ^{\infty of any the event that is defined in terms of values}the of probability X may be Given joint density or distribution function, \left| e^{j\omega x} \right| \ y) is found. P{X ≤in x, Y ≤ = FofX,Y of any For eventexample, that is defined terms of y} values X(x, may be \left| f_X(x) \right| dxas in Eq. (5). found by integrating the joint density function found. For example, P{X ≤ x, Y ≤ y} = FX,Y (x, y) isdx = 1 = \int _{-\infty } ^{\infty } f_X(x) Two important properties of the joint density function are found by integrating the joint density function as in Eq. (5). \end{eqnarray*} given below. Two important properties of the joint density function are given X 1) fbelow. XY (x, y) ≥ 0 for all x, y. Z −λx ∞ Z ∞ (x) = λe u(x) 1) fXY (x, y) ≥ 0 for all x, y. thenM.Hayes the (CAU-GT) characteristicfX function is November 08, 2011 30 / 54 fXY (x, y)dxdy = 1 Z ∞ (CAU-GT) Z ∞ Z ∞ November 08, 2011 29 / 54 2) Z ∞ M.Hayes −∞ −∞ then the characteristic function is 2) fXY (x, y)dxdy = 1 MX (jω) = Z ∞λejωx e−λx dx = λ Z ∞e(jω−λ)x dx The first property follows from the monotonicity of the −∞ −∞ 0 (jω−λ)x MX (jω) = 0 λejωx e−λxdx = λ e dx distribution Specifically, FX (x) is aofnonThe LAfirst property follows from since the monotonicity the λ λ Figures TEX function. ∞ 0 0 = e(jω−λ)x ∞= decreasing function of x, then the derivative will always be distribution function. Specifically, since FX (x) is a nonjω − λ −λjω λ λ (jω−λ)x0 = e = non-negative for all ofx. x,The propertywill follows from decreasing function thensecond the derivative always be jω − λ function of 0 ω. λ − jω \subsection{Gaussian Random Variable} which is a complex-valued Eq. (5) by letting x → ∞ and y → ∞, non-negative for all x. The second property follows from which is a complex-valued function of ω. Z ∞xGaussian Z ∞ zero-mean random $X$ has a density Eq. (5)A by letting → ∞ and y →variable ∞, B. Gaussian Random Variable function Z of ∞ the Z form ∞ fXY (x, y)dxdy = 1 \[ f_X(x) \frac {1}{\sigma _x\sqrt {2\pi }} B.A Gaussian Variable −∞ =−∞ zero-meanRandom Gaussian random variable X has a density fXY (x, y)dxdy =1 e^{-x^2/2\sigma _x^2} \] function of the form −∞ −∞ A zero-mean Gaussian random variable X has a density As was the$\sigma case for_x^2$ a single random variable, it is important where is the variance of $X$. function of the form 2 2 1 A plot the density a Gaussian to As understand that the jointfunction density (x,several y) is was the of case for a single randomoffunction variable, itfXY is for important fX (x) = √ e−x /2σx different values of $\sigma _x$ is shown in 2 2 1 atoprobability andjoint not density a probability. Probabilities understanddensity that the function fXY (x, y)are is Fig.~\ref{fig:Gaussian}. fX (x) =σx √2π e−x /2σx by integrating function. For example, the a probability density the and density not a probability. Probabilities are σx A2πplot of the density function found \begin{figure} where σx2 is the variance of X. probability of the eventthe density function. For example, the found \begin{center} by integrating ofwhere a Gaussian forvariance several of different values of σdensity in σx2 is the X. A plot of the function x is shown \includegraphics[width=\hsize]{images/Gaussian.png}\\ probability of the event Fig. of a1.Gaussian for several different values of σx is shown in A = {x1 ≤ X ≤ x2 } ∪ {y1 ≤ X ≤ y2 } \end{center} \caption{A Function} The1.characteristic function is Fig. A = {x1 Gaussian ≤ X ≤ xDensity 2 } ∪ {y1 ≤ X ≤ y2 } Z \label{fig:Gaussian} is found by integrating the joint density function as follows ∞ The characteristic is −x2 /2σ2 1 function jωx \end{figure} Z x2 function Z y2 x Z ∞e e is found by integrating the joint density as follows MX (jω) = √ 2 2 1 jωx −x /2σ σ 2π −∞ e x P{x1 ≤ X M.Hayes ≤ x2 ,(CAU-GT) y1 ≤ X ≤ y2 } = Z x2 Z y2 fXY (x, y)dxdy M (jω) = x √ Ze ∞ November 08, 2011 32 / 54 M.HayesX(CAU-GT) November 08, 2011 31 / 54 σ 2π 2 2 2 2 1 x σ /2 −∞ P{x1 ≤ X ≤ x2 , y1 ≤ X ≤ y2 } = x1 y1 fXY (x, y)dxdy −ω −(x−jωσx ) /2σx x Z function of the form Given the joint density or distribution function, the probability 2 1 ys −x2 /2σx √ f (x) = e of any event that isX defined in terms of values of X may be Underbraces σx 2π found. For example, P{X ≤ x, Y ≤ y} = FX,Y (x, y) is 2 where σ is the variance of X. A plot of the density found byx integrating the joint density function as infunction Eq. (5). of a Gaussian for several different values of σ is shown x function in Two important properties of the joint density are λ, Fig. 1. given below. The characteristic function is 1) fXY (x, y) ≥ 10 forZ all ∞ x, y. 2 2 ∞ Z = ∞ √ ejωx e−x /2σx MZX (jω) σfxXY2π −∞ 2) (x, y)dxdy = 1 Z ∞ −∞ −∞ 2 2 2 1 −ω σx /2 −(x−jωσx )2 /2σx √ = e e The first property follows from−∞the monotonicitydxof the σx 2π | {z FX (x) is a} nondistribution function. Specifically, since ity =1 decreasing function of x, then the derivative will always be non-negative for all x. The second property follows from Eq. (5) by letting x → ∞ and y → ∞, Z ∞Z ∞ fXY (x, y)dxdy = 1 M.Hayes (CAU-GT) −∞ −∞ November 08, 2011 As was the case for a single random variable, it is important to understand that the joint density function fXY (x, y) is LATEXdensity and not a probability. Probabilities a probability are found by integrating the density function. For example, the probability of the event = {x1 ≤ X ≤ function x2 } ∪ {y1is≤ X ≤ y2 } TheA characteristic \begin{align*} is found by M_X(j\omega integrating the ) joint density function as follows Z x2 Z y{2\pi &= \frac {1}{\sigma _x\sqrt }} 2 \int _{-\infty } ^{\infty } e^{j\omega x} fXY (x, y)dxdy P{x1 ≤ X ≤ x2 , y1 ≤ X ≤ y2 } = e^{-x^2/2\sigma _x^2} \\ y1 x1 (6) &= e^{-\omega ^2 \sigma _x^2/2} \underbrace{\frac _x\sqrt and to find the probability that (x, y){1}{\sigma ∈ R where R is{2\pi an }} \int _{-\infty } ^{\infty } arbitrarily defined region in the x-y plane, we integrate the e^{-(x - j\omega \sigma _x)^2/2\sigma _x^2} dx}_{=1} \end{align*} 33 / 54 As was the case for a single random variable, it is important to understand that the joint density function fXY (x, y) is a probability density and not a probability. Probabilities are Double Integrals found by integrating the density function. For example, the on probability of the event in A = {x1 ≤ X ≤ x2 } ∪ {y1 ≤ X ≤ y2 } is found by integrating the joint density function as follows Z x2 Z y2 P{x1 ≤ X ≤ x2 , y1 ≤ X ≤ y2 } = fXY (x, y)dxdy x1 y1 (6) and to find the probability that (x, y) ∈ R where R is an arbitrarily defined region in the x-y plane, we integrate the M.Hayes (CAU-GT) November 08, 2011 35 / 54 M.Hayes (CAU-GT) November 08, 2011 34 / 54 LATEX \begin{equation} \pr \{ x_1 \leq X \leq x_2, \ y_1 \leq X \leq y_2 \} = \int _{x_1}^{x_2} \int _{y_1}^{y_2} f_{XY}(x, y) dx dy \label{eq:IntegrateJointDensity} \end{equation} M.Hayes (CAU-GT) November 08, 2011 36 / 54 4 IEEE TRANSACTIONS ON LATEX LATEX Integrals with Limits at Bottom of Integral Sign density function over R, 4 P{(X, Y ) ∈ R} = ZZ fXY (x, y)dxdy IEEE TRANSACTIONS ON LATEX (7) R density function over R, Unlike the joint distribution function, which is constrained ZZ to have values between zero one, is no such P{(X, Y ) ∈ R} = andfXY (x,there y)dxdy (7) constraint on the joint density function. In fact, fXY (x, y) may be arbitrarily large as longRas the integral of fXY (x, y) the joint distribution overUnlike all x and y is equal to one function, which is constrained to have values between zero and one, there is no such constraint on the joint density function. In M fact, fXY (x, y) VII. C ORRELATION AND C ORRELATION ATRICES may large as variables, long as the of fensemble XY (x, y) For be twoarbitrarily or more random anintegral important over allisxthe andcorrelation. y is equal Given to onetwo random variables X and average Y , the correlation is defined by VII. C ORRELATION AND C ORRELATION M ATRICES M.Hayes (CAU-GT) November 08, 2011 37 / 54 E{X1 X2 }an important ensemble X1 X2 =variables, For two or more rrandom is the correlation. Given two randomtovariables X and Ifaverage we consider these two random variables be a random Y , the correlation is defined by vector Matrices T X X= [X 1 , X2 ] rX 1 2 = E{X1 X2 } T then theconsider correlation matrix the expected value of XX , If we these two is random variables to be a random " # 2 vector E{X } E{X X } 1 2 1 RX = E{XXT } = X = [X1 , X2 ]T E{X2 X1 } E{X22 } then the correlation matrix is the expected value of XXT , For n random variables, the"correlation matrix has the #form 1 X2 } E{X12 } E{X T r11 r · · · r1n 12 RX = E{XX } = r21 rE{X 22 } ·X · · 1 } r2n E{X 2 22 RX = . .. .. . . . . the correlation For n random variables, . matrix . . has the form r n1 r11 rn2 r12 · ·· ·· · rnn r1n r21 r22 · · · r2n where RX =r = .. E{X .. X. }. .. kl . . k l . . rn1 estimated rn2 · · · from rnn a set of realThese correlations are often M.Hayes (CAU-GT) November 08, 2011 39 / 54 izations where (experimental outcomes) of the random variables as \begin{equation} \pr \{ (X,Y) \in R \} = \iint\limits _{R} f_{XY}(x,y) dx dy \label{eq:IntegrateOverR} \end{equation} M.Hayes (CAU-GT) November 08, 2011 38 / 54 LATEX ${\bf X}{\bf X}^T$, \[ {\bf R}_X = E\{ {\bf X} {\bf X}^T \} = \left[ \begin{array}{cc} E\{ X_1^2 \} & E\{ X_1X_2 \} \\[1ex] E\{ X_2X_1\} & E\{ X_2^2 \} \end{array} \right] \] For $n$ random variables, the correlation matrix has the form \[ {\bf R}_X = \begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1n} \\ r_{21} & r_{22} & \cdots & r_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ r_{n1} & r_{n2} & \cdots & r_{nn} \\ \end{bmatrix} \] M.Hayes (CAU-GT) November 08, 2011 40 / 54 (7) ned uch y) y) ble nd rn1 rn2 where ··· rnn LATEX Sums and Integrals in rFractions kl = E{Xk Xl } These correlations are often estimated from a set of realizations (experimental outcomes) of the random variables as follows Sample cross-correlation r̂xy = " n X (xi − x)(yi − y) i=1 n X i=1 (xi − x)2 n X i=1 (yi − y)2 \[ \hat{r}_{xy} = \frac {\displaystyle \sum _{i=1}^n (x_i - x)(y_i - y)} {\displaystyle \left[ \sum _{i=1}^n (x_i - x)^2 \sum _{i=1}^n (y_i - y)^2 \right]} \] # where r̂xy is called a sample autocorrelation. VIII. C ONCLUSION There are many excellent textbooks where the reader may find advanced developments of the results presented in this M.Hayes (CAU-GT) 08, 2011 41 / 54 paper. The classic work in the field is the text byNovember Papoulis [1]. Another recommended text is [2]. An introduction to Monte Carlo simulations may be found in [3]. A The Subpicture Environment IEEE TRANSACTIONS ON LTEX M.Hayes (CAU-GT) November 08, 2011 42 / 54 November 08, 2011 44 / 54 LATEX R EFERENCES [1] A. Papoulis and S. Pillai, Probability, Random Variables, and Stochasic Processes. New York: McGraw-Hill, 2002. [2] H. Larsen and B. Shubert, Probabilistic Models in Engineering Sciences, Vol. 1. New York: John Wiley and Sons, 1979. [3] S. Raychaudhuri, “Introduction to monte carlo simulation,” in Simulation Conference, 2008. WSC 2008. Winter, pp. 91 –100, Dec. 2008. (a) Density Function Fig. 2. (b) Distribution Function \begin{figure} \begin{center} \subfigure[Density Function]{ \includegraphics[width=0.45\hsize] {images/Chi-Square_distributionPDF.png}} \subfigure[Distribution Function]{ \includegraphics[width=0.45\hsize] {images/Chi-Square_distributioncDF.png}} \end{center} \caption{The Chi-square Random Variable.} \label{fig:ChiSquare} \end{figure} The Chi-square Random Variable. M.Hayes (CAU-GT) November 08, 2011 IX. C ONCLUSION 43 / 54 M.Hayes (CAU-GT) i=1 (xi − x)2 i=1 (yi − y)2 LATEX References Using BibTEX where r̂xy is called a sample autocorrelation. VIII. C ONCLUSION \section{Conclusion} There are many excellent textbooks where the reader may find advanced developments of the results presented in this paper. The classic work in the field is the text by Papoulis [1]. Another recommended text is [2]. An introduction to Monte Carlo simulations may be found in [3]. R EFERENCES [1] A. Papoulis and S. Pillai, Probability, Random Variables, and Stochasic Processes. New York: McGraw-Hill, 2002. [2] H. Larsen and B. Shubert, Probabilistic Models in Engineering Sciences, Vol. 1. New York: John Wiley and Sons, 1979. [3] S. Raychaudhuri, “Introduction to monte carlo simulation,” in Simulation Conference, 2008. WSC 2008. Winter, pp. 91 –100, Dec. 2008. M.Hayes (CAU-GT) November 08, 2011 45 / 54 BibTEX 2 3 \bibliography{mybibliography} \bibliographystyle{ieeetr} \end{document} M.Hayes (CAU-GT) November 08, 2011 46 / 54 BibTEX File BibTEX makes it easy to cite sources in a consistent manner, by separating bibliographic information from the presentation of this information. BibTEX takes, as input 1 There are many excellent textbooks where the reader may find advanced developments of the results presented in this paper. The classic work in the field is the text by Papoulis~\cite{Pap2002}. Another recommended text is~\cite{Larsen}. An introduction to Monte Carlo simulations may be found in~\cite{MonteCarlo}. An .aux file produced by LATEX on an earlier run; A .bst file (the style file), that specifies the general reference-list style and specifies how to format individual entries, A .bib file(s) constituting a database of all reference-list entries the user might ever hope to use. BibTEX chooses from the .bib file(s) only those entries specified by the .aux file (that is, those given by LATEX’s \cite or \nocite commands), and creates as output a .bbl file containing these entries together with the formatting commands specified by the .bst file. LATEX uses the .bbl file, perhaps edited by the user, to produce the reference list. M.Hayes (CAU-GT) November 08, 2011 47 / 54 @book{Pap2002, author = {A. Papoulis and S. Pillai}, title = {Probability, Random Variables, and Stochasic Processes}, publisher = {McGraw-Hill}, Address = {New York}, year = {2002} } @INPROCEEDINGS{MonteCarlo, author={Raychaudhuri, S.}, booktitle={Simulation Conference, 2008. WSC 2008. Winter}, title={Introduction to Monte Carlo Simulation}, year={2008}, month={Dec.}, volume={}, number={}, pages={91 -100}, keywords={Monte Carlo simulation;repeated random sampling; statistical analysis;Monte Carlo methods;random processes}, doi={10.1109/WSC.2008.4736059}, ISSN={},} M.Hayes (CAU-GT) November 08, 2011 48 / 54 Manual Specification of References BibTEX and IEEExplore \begin{thebibliography}{9} \bibitem{Pap2002} A. Papoulis and S. Pillai, \emph{Probability, Random Variables, and Stochastic Processes}, Mc-Graw Hill, 2002. \bibitem{Larsen} H Larsen and B. Shubert, \emph{Probabilistic Models in Engineering Sciences, Vol. 1}, John Wiley and Sons, New York, 1979. \bibitem{MonteCarlo} S. Raychaudhuri, "Introduction to Monte Carlo Simulation," \emph{Simulation Conference}, pp. 91-100, Dec. 2008 IEEExplore and other databases will export citations into BibTEX format, as well as others. \end{thebibliography} M.Hayes (CAU-GT) November 08, 2011 49 / 54 Ifthen Package M.Hayes (CAU-GT) November 08, 2011 50 / 54 Verbatim Text To include computer listings or other similar text, we would like to have unformatted text to produce something like: \begin{align*} M_X(j\omega ) &= \frac {1}{\sigma _x\sqrt {2\pi }} \int _{-\infty } ^{\infty } e^{j\omega x} e^{-x^2/2\sigma _x^2} \\ &= e^{-\omega ^2 \sigma _x^2/2} \underbrace{\frac {1}{\sigma _x\sqrt {2\pi }} \int _{-\infty } ^{\infty } e^{-(x - j\omega \sigma _x)^2/2\sigma _x^2} dx}_{=1} \end{align*} %---------Ignore all text from here until \fi --------%---Replace \iffalse with \iftrue to include text ----\iffalse It is clear that this probability assignment satisfies the first probability axiom since all probabilities in Eq.~\ref{eq:prob_assgn} are positive. \fi %------------end ---------------- There are several ways to introduce text that won’t be interpreted by the compiler. I I M.Hayes (CAU-GT) November 08, 2011 51 / 54 With the verbatim environment, everything input between a \begin{verbatim} and an \end{verbatim} command will be processed as if by a typewriter. Also see the \verb command for short in-line verbatim text. M.Hayes (CAU-GT) November 08, 2011 52 / 54 Graphics Next Time Now that you have your beautifully typeset journal paper or article, you want your figures, block diagrams, plots and other graphics to be beautifully typset. There are a number of very powerful packages that allow you to create graphics in postscript or PDF file format. Some of these are: I I I I I I How to prepare and deliver an effective presentation. Presentations with PowerPoint Using Aurora xfig, TikZ and PGF, XY-Pic, PSTricks and PDFTricks, Metapost Adobe Illustrator LATEX Presentations using the Beamer and Prosper Classes See the web for a description of these packages and for documentation. M.Hayes (CAU-GT) November 08, 2011 53 / 54 M.Hayes (CAU-GT) November 08, 2011 54 / 54