DOCSLIB.ORG

2.4 Formal Power Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2.4 Formal Power Series 52 CHAPTER 2. ADVANCED COUNTING AND GENERATING FUNCTIONS 2.4 Formal Power Series In this chapter, we will first try to develop the theory of generating functions by getting closed form expressions for some known recurrence relations. These ideas will be used later to get some binomial identities. To do so, we first recall from Page 41 that for all n Q and k Z, k 0, the binomial ∈ n(n 1)(∈n 2) ≥ (n k + 1) coefficients, n , are well defined, using the idea that n = − − · · · − . We k k k! now start with the definition of “formal power series” over Q and study its properties in some detail. n Definition 2.4.1 (Formal power series). An algebraic expression of the form f(x)= anx , n≥0 where a Q for all n 0, is called a formal power series in the indeterminate x overPQ. n ∈ ≥ The set of all formal power series in the indeterminate x, with coefficients from Q will be denoted by (x). P Remark 2.4.2. 1. Given a sequence of numbers an Q : n = 0, 1, 2,... , one associates { ∈ n } n x n two formal power series, namely, anx and an . The expression anx is called n≥0 n≥0 n! n≥0 P P xn P the generating function and the expression an is called the exponential generating n≥0 n! function, for the numbers a : n 0 . P { n ≥ } n n 2. Let f(x) = anx be a formal power series. Then the coefficient of x , for n 0, in n≥0 ≥ f(x) is denotedP by [xn]f(x). That is, a = [x0]f(x) and a = [xn]f(x), for n 1. 0 n ≥ n 3. One thinks of anx as an algebraic expression. In general, one is interested only in n≥0 computing the coefficientP of certain power of x and not in evaluating them for any value of x. But, if at all there is a need to evaluate it at a point, say x0, then one needs to determine its “radius of convergence” and then evaluate it if x0 lies within that radius. We need the following definition to proceed further. n Definition 2.4.3 (Equality of two formal power series). Two elements f(x) = anx and n≥0 n g(x)= bnx of (x) are said to be equal if an = bn, for all n 0. P n≥0 P ≥ P We are now ready to define the algebraic rules: n n Definition 2.4.4. Let f(x)= anx , g(x)= bnx (x). Then their n≥0 n≥0 ∈ P P P 1. sum/addition is defined by n n n f(x)+ g(x)= anx + bnx = (an + bn)x . nX≥0 nX≥0 Xn≥0 2.4. FORMAL POWER SERIES 53 2. product is defined by n f(x) g(x)= a xn b xn = c xn, where c = a b , for n 0. · n · n n n k n−k ≥ Xn≥0 nX≥0 nX≥0 Xk=0 This product is also called the Cauchy product. xn Remark 2.4.5. 1. In case of exponential power series, the product of f(x)= an and n≥0 n! n n n x x n P g(x)= bn equals dn , where dn = k akbn−k, for n 0. n≥0 n! n≥0 n! k=0 ≥ P P x P 2. Note that the expression ee −1 is a well defined formal power series as the definition ey = n x n y x (e 1) implies that ee −1 = − and hence n≥0 n! n≥0 n! P P x n m x n x (e 1) (e 1) [xm]ee −1 = [xm] − = [xm] − . (2.1) n! n! n=0 nX≥0 X x That is, for each m 0, [xm]ee −1 is a sum of a finite number of real numbers. Where x≥ x as the expression ee is not a formal power series as the computation of [xm]ee , for all m 0, will indeed require an infinite sum. ≥ Thus, under the algebraic operations defined above, it can be checked that the set (x) forms P a Commutative Ring with identity, where the identity element is given by the formal power series n f(x) = 1. In this ring, the element f(x) = anx is said to have a reciprocal if there exists n≥0 n another element g(x)= bnx (x) suchP that f(x) g(x) = 1. So, the question arises, under n≥0 ∈ P · what conditions on the coefficientsP of f(x), can we find g(x) (x) such that f(x)g(x) = 1. ∈ P The answer to this question is given in the following proposition. n Proposition 2.4.6. Let f(x) = anx (x). Then there exists g(x) (x) satisfying n≥0 ∈ P ∈ P f(x) g(x) = 1 if and only if a = 0P. · 0 6 n Proof. Let g(x)= bnx (x). Then, by the definition of Cauchy product, f(x)g(x)= n≥0 ∈ P n n P cnx , where cn = akbn−k, for all n 0. Therefore, using the definition of equality of two n≥0 k=0 ≥ powerP series, we see thatP f(x)g(x) = 1 if and only if c = 1 and c = 0, for all n 1. 0 n ≥ Therefore, if a0 = 0 then c0 = 0 and hence the Cauchy product f(x)g(x) can never equal 1. However, if a = 0, then the coefficients b ’s can be recursively obtained as follows: 0 6 n 1 b0 = as 1= c0 = a0b0. a0 1 b1 = − (a1b0)as0= c1 = a0b1 + a1b0. a0 · 54 CHAPTER 2. ADVANCED COUNTING AND GENERATING FUNCTIONS 1 b2 = − (a2b0 + a1b1)as 0= c2 = a0b2 + a1b1 + a2b0. And in general, if we have already a0 · computed the values of b , for k r, then k ≤ 1 br+1 = − (ar+1b0 + arb1 + + a1br)as0= cr+1 = ar+1b0 + arb1 + + a1br + a0br+1. a0 · · · · · · · Note that if the coefficients a ’s come from , a commutative ring with unity, then one needs n R b to be an invertible element of for Proposition 2.4.6 to hold true. Let us now look at the 0 R n n composition of two formal power series. Recall that, if f(x)= anx , g(x)= bnx (x) n≥0 n≥0 ∈ P n m n then the composition (f g)(x) = f(g(x)) = an(g(x)) =P an( bmx P) may not be ◦ n≥0 n≥0 m≥0 defined (just to compute the constant term ofP the compositionP, one mayP have to look at an infinite sum). For example, let f(x) = ex and g(x) = x + 1. Note that g(0) = 1 = 0. Here, 6 (f g)(x) = f(g(x)) = f(x +1) = ex+1. So, as function f g is well defined, but there is no ◦ ◦ x+1 k x+1 formal procedure to write e as akx (x) (i.e., with ak Q) and hence e is not a k≥0 ∈ P ∈ formal power series over Q. P The next result gives the condition under which the composition (f g)(x) is well defined. ◦ n n Proposition 2.4.7. Let f(x)= anx and g(x)= bnx be two formal power series. Then n≥0 n≥0 the composition (f g)(x) is wellP defined if either f isP a polynomial or b = 0. ◦ 0 n Moreover, suppose that a0 = 0. Then, there exists g(x) = bnx , with b0 = 0, such that n≥0 (f g)(x)= x. Furthermore, (g f)(x) is well defined and (g Pf)(x)= x. ◦ ◦ ◦ n Proof. Let (f g)(x) = f(g(x)) = cnx and suppose that either f is a polynomial or ◦ n≥0 b = 0. Then to compute c = [xk] (f Pg)(x), for k 0, one just needs to consider the terms 0 k ◦ ≥ a + a g(x)+ a (g(x))2 + + a (g(x))k. Hence, each c is a real number and (f g)(x) is well 0 1 2 · · · k k ◦ defined. This completes the proof of the first portion. The proof of the other part is left to the readers. We now define the formal differentiation of elements of (x) and state a result without proof. P n Definition 2.4.8 (Differentiation). Let f(x) = anx (x). Then the formal differentia- n≥0 ∈ P tion of f(x), denoted Df(x), is defined by P Df(x)= a + 2a x + + na xn−1 + = na xn−1. 1 2 · · · n · · · n nX≥1 n Proposition 2.4.9. Let f(x) = anx (x). Then f(x) = a0, a constant, whenever n≥0 ∈ P x Df(x) = 0. Also, f(x)= a0e wheneverP Df(x)= f(x)..
Load more

Recommended publications
  • The Unilateral Z–Transform and Generating Functions
    The Unilateral z{Transform and Generating Functions Recall from \Discrete{Time Linear, Time Invariant Systems and z-Transforms" that the behaviour of a discrete{time LTI system is determined by its impulse response function h[n] and that the z{transform of h[n] is 1 k H(z) = z− h[k] k=X −∞ If the LTI system is causal, then h[n] = 0 for all n < 0 and 1 k H(z) = z− h[k] Xk=0 Definition 1 (Unilateral z{Transform) The unilateral z{transform of the discrete{time signal x[n] (whether or not x[n] = 0 for negative n's) is defined to be 1 n (z) = z− x[n] X nX=0 When there is any danger of confusing the regular z{transform with the unilateral z{transform, 1 n X(z) = z− x[n] nX= 1 − is called the bilateral z{transform. Example 2 The signal x[n] = anu[n] is zero for all n < 0. So the unilateral z{transform of x[n] is the same as the ordinary z{transform. So, as we saw in Example 7 of \Discrete{Time Linear, Time Invariant Systems and z-Transforms", 1 n n 1 (z) = X(z) = z− a = 1 X 1 z− a nX=0 − 1 provided that z− a < 1, or equivalently z > a . Since the unilateral z{transform of any x[n] is always j j j j j j equal to the bilateral z{transform of the right{sided signal x[n]u[n], the region of convergence of a unilateral z{transform is always the exterior of a circle.
    [Show full text]
  • Ordinary Generating Functions
    CHAPTER 10 Ordinary Generating Functions Introduction We’ll begin this chapter by introducing the notion of ordinary generating functions and discussing the basic techniques for manipulating them. These techniques are merely restatements and simple applications of things you learned in algebra and calculus. You must master these basic ideas before reading further. In Section 2, we apply generating functions to the solution of simple recursions. This requires no new concepts, but provides practice manipulating generating functions. In Section 3, we return to the manipulation of generating functions, introducing slightly more advanced methods than those in Section 1. If you found the material in Section 1 easy, you can skim Sections 2 and 3. If you had some difficulty with Section 1, those sections will give you additional practice developing your ability to manipulate generating functions. Section 4 is the heart of this chapter. In it we study the Rules of Sum and Product for ordinary generating functions. Suppose that we are given a combinatorial description of the construction of some structures we wish to count. These two rules often allow us to write down an equation for the generating function directly from this combinatorial description. Without such tools, we may get bogged down in lengthy algebraic manipulations. 10.1 What are Generating Functions? In this section, we introduce the idea of ordinary generating functions and look at some ways to manipulate them. This material is essential for understanding later material on generating functions. Be sure to work the exercises in this section before reading later sections! Definition 10.1 Ordinary generating function (OGF) Suppose we are given a sequence a0,a1,..
    [Show full text]
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • [Math.CA] 1 Jul 1992 Napiain.Freape Ecnlet Can We Example, for Applications
    Convolution Polynomials Donald E. Knuth Computer Science Department Stanford, California 94305–2140 Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F (x), F (x), F (x),... forms a convolution family if F (x) has degree n 0 1 2 n ≤ and if the convolution condition F (x + y)= F (x)F (y)+ F − (x)F (y)+ + F (x)F − (y)+ F (x)F (y) n n 0 n 1 1 · · · 1 n 1 0 n holds for all x and y and for all n 0. Many such families are known, and they appear frequently ≥ n in applications. For example, we can let Fn(x)= x /n!; the condition (x + y)n n xk yn−k = n! k! (n k)! kX=0 − is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial x coefficient n ; the corresponding identity n x + y x y = n k n k Xk=0 − is commonly called Vandermonde’s convolution. How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 n 4: ≤ ≤ F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! arXiv:math/9207221v1 [math.CA] 1 Jul 1992 conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}] ==Series[Sum[F[k,x]F[n-k,y],{k,0,n}],{x,0,n},{y,0,n}]] Solve[Table[conv[n],{n,0,4}], [Flatten[Table[f[i,j],{i,0,4},{j,0,4}]]]] Mathematica replies that the F ’s are either identically zero or the coefficients of Fn(x) = fn0 + f x + f x2 + + f xn /n! satisfy n1 n2 · · · nn f00 = 1 , f10 = f20 = f30 = f40 = 0 , 2 3 f22 = f11 , f32 = 3f11f21 , f33 = f11 , 2 2 4 f42 = 4f11f31 + 3f21 , f43 = 6f11f21 , f44 = f11 .
    [Show full text]
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • The Discovery of the Series Formula for Π by Leibniz, Gregory and Nilakantha Author(S): Ranjan Roy Source: Mathematics Magazine, Vol
    The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha Author(s): Ranjan Roy Source: Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690896 Accessed: 27-02-2017 22:02 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine This content downloaded from 195.251.161.31 on Mon, 27 Feb 2017 22:02:42 UTC All use subject to http://about.jstor.org/terms ARTICLES The Discovery of the Series Formula for 7r by Leibniz, Gregory and Nilakantha RANJAN ROY Beloit College Beloit, WI 53511 1. Introduction The formula for -r mentioned in the title of this article is 4 3 57 . (1) One simple and well-known moderm proof goes as follows: x I arctan x = | 1 +2 dt x3 +5 - +2n + 1 x t2n+2 + -w3 - +(-I)rl2n+1 +(-I)l?lf dt. The last integral tends to zero if Ix < 1, for 'o t+2dt < jt dt - iX2n+3 20 as n oo.
    [Show full text]
  • AN INTRODUCTION to POWER SERIES a Finite Sum of the Form A0
    AN INTRODUCTION TO POWER SERIES PO-LAM YUNG A finite sum of the form n a0 + a1x + ··· + anx (where a0; : : : ; an are constants) is called a polynomial of degree n in x. One may wonder what happens if we allow an infinite number of terms instead. This leads to the study of what is called a power series, as follows. Definition 1. Given a point c 2 R, and a sequence of (real or complex) numbers a0; a1;:::; one can form a power series centered at c: 2 a0 + a1(x − c) + a2(x − c) + :::; which is also written as 1 X k ak(x − c) : k=0 For example, the following are all power series centered at 0: 1 x x2 x3 X xk (1) 1 + + + + ::: = ; 1! 2! 3! k! k=0 1 X (2) 1 + x + x2 + x3 + ::: = xk: k=0 We want to think of a power series as a function of x. Thus we are led to study carefully the convergence of such a series. Recall that an infinite series of numbers is said to converge, if the sequence given by the sum of the first N terms converges as N tends to infinity. In particular, given a real number x, the series 1 X k ak(x − c) k=0 converges, if and only if N X k lim ak(x − c) N!1 k=0 exists. A power series centered at c will surely converge at x = c (because one is just summing a bunch of zeroes then), but there is no guarantee that the series will converge for any other values x.
    [Show full text]
  • The Bloch-Wigner-Ramakrishnan Polylogarithm Function
    Math. Ann. 286, 613424 (1990) Springer-Verlag 1990 The Bloch-Wigner-Ramakrishnan polylogarithm function Don Zagier Max-Planck-Insfitut fiir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany To Hans Grauert The polylogarithm function co ~n appears in many parts of mathematics and has an extensive literature [2]. It can be analytically extended to the cut plane ~\[1, ~) by defining Lira(x) inductively as x [ Li m_ l(z)z-tdz but then has a discontinuity as x crosses the cut. However, for 0 m = 2 the modified function O(x) = ~(Liz(x)) + arg(1 -- x) loglxl extends (real-) analytically to the entire complex plane except for the points x=0 and x= 1 where it is continuous but not analytic. This modified dilogarithm function, introduced by Wigner and Bloch [1], has many beautiful properties. In particular, its values at algebraic argument suffice to express in closed form the volumes of arbitrary hyperbolic 3-manifolds and the values at s= 2 of the Dedekind zeta functions of arbitrary number fields (cf. [6] and the expository article [7]). It is therefore natural to ask for similar real-analytic and single-valued modification of the higher polylogarithm functions Li,. Such a function Dm was constructed, and shown to satisfy a functional equation relating D=(x-t) and D~(x), by Ramakrishnan E3]. His construction, which involved monodromy arguments for certain nilpotent subgroups of GLm(C), is completely explicit, but he does not actually give a formula for Dm in terms of the polylogarithm. In this note we write down such a formula and give a direct proof of the one-valuedness and functional equation.
    [Show full text]
  • 5. Polynomial Rings Let R Be a Commutative Ring. a Polynomial of Degree N in an Indeterminate (Or Variable) X with Coefficients
    5. polynomial rings Let R be a commutative ring. A polynomial of degree n in an indeterminate (or variable) x with coefficients in R is an expression of the form f = f(x)=a + a x + + a xn, 0 1 ··· n where a0, ,an R and an =0.Wesaythata0, ,an are the coefficients of f and n ··· ∈ ̸ ··· that anx is the highest degree term of f. A polynomial is determined by its coeffiecients. m i n i Two polynomials f = i=0 aix and g = i=1 bix are equal if m = n and ai = bi for i =0, 1, ,n. ! ! Degree··· of a polynomial is a non-negative integer. The polynomials of degree zero are just the elements of R, these are called constant polynomials, or simply constants. Let R[x] denote the set of all polynomials in the variable x with coefficients in R. The addition and 2 multiplication of polynomials are defined in the usual manner: If f(x)=a0 + a1x + a2x + a x3 + and g(x)=b + b x + b x2 + b x3 + are two elements of R[x], then their sum is 3 ··· 0 1 2 3 ··· f(x)+g(x)=(a + b )+(a + b )x +(a + b )x2 +(a + b )x3 + 0 0 1 1 2 2 3 3 ··· and their product is defined by f(x) g(x)=a b +(a b + a b )x +(a b + a b + a b )x2 +(a b + a b + a b + a b )x3 + · 0 0 0 1 1 0 0 2 1 1 2 0 0 3 1 2 2 1 3 0 ··· Let 1 denote the constant polynomial 1 and 0 denote the constant polynomial zero.
    [Show full text]
  • Skew and Infinitary Formal Power Series
    Appeared in: ICALP’03, c Springer Lecture Notes in Computer Science, vol. 2719, pages 426-438. Skew and infinitary formal power series Manfred Droste and Dietrich Kuske⋆ Institut f¨ur Algebra, Technische Universit¨at Dresden, D-01062 Dresden, Germany {droste,kuske}@math.tu-dresden.de Abstract. We investigate finite-state systems with costs. Departing from classi- cal theory, in this paper the cost of an action does not only depend on the state of the system, but also on the time when it is executed. We first characterize the terminating behaviors of such systems in terms of rational formal power series. This generalizes a classical result of Sch¨utzenberger. Using the previous results, we also deal with nonterminating behaviors and their costs. This includes an extension of the B¨uchi-acceptance condition from finite automata to weighted automata and provides a characterization of these nonter- minating behaviors in terms of ω-rational formal power series. This generalizes a classical theorem of B¨uchi. 1 Introduction In automata theory, Kleene’s fundamental theorem [17] on the coincidence of regular and rational languages has been extended in several directions. Sch¨utzenberger [26] showed that the formal power series (cost functions) associated with weighted finite automata over words and an arbitrary semiring for the weights, are precisely the rational formal power series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari [6], Hafner [14], Katritzke [16], Jiang, Litow and de Vel [15]) and in speech-to-text processing (Mohri [20], [21], Buchsbaum, Giancarlo and Westbrook [4]).
    [Show full text]
  • Euler and His Work on Infinite Series
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously.
    [Show full text]
  • Functions of Random Variables
    Names for Eg(X ) Generating Functions Topic 8 The Expected Value Functions of Random Variables 1 / 19 Names for Eg(X ) Generating Functions Outline Names for Eg(X ) Means Moments Factorial Moments Variance and Standard Deviation Generating Functions 2 / 19 Names for Eg(X ) Generating Functions Means If g(x) = x, then µX = EX is called variously the distributional mean, and the first moment. • Sn, the number of successes in n Bernoulli trials with success parameter p, has mean np. • The mean of a geometric random variable with parameter p is (1 − p)=p . • The mean of an exponential random variable with parameter β is1 /β. • A standard normal random variable has mean0. Exercise. Find the mean of a Pareto random variable. Z 1 Z 1 βαβ Z 1 αββ 1 αβ xf (x) dx = x dx = βαβ x−β dx = x1−β = ; β > 1 x β+1 −∞ α x α 1 − β α β − 1 3 / 19 Names for Eg(X ) Generating Functions Moments In analogy to a similar concept in physics, EX m is called the m-th moment. The second moment in physics is associated to the moment of inertia. • If X is a Bernoulli random variable, then X = X m. Thus, EX m = EX = p. • For a uniform random variable on [0; 1], the m-th moment is R 1 m 0 x dx = 1=(m + 1). • The third moment for Z, a standard normal random, is0. The fourth moment, 1 Z 1 z2 1 z2 1 4 4 3 EZ = p z exp − dz = −p z exp − 2π −∞ 2 2π 2 −∞ 1 Z 1 z2 +p 3z2 exp − dz = 3EZ 2 = 3 2π −∞ 2 3 z2 u(z) = z v(t) = − exp − 2 0 2 0 z2 u (z) = 3z v (t) = z exp − 2 4 / 19 Names for Eg(X ) Generating Functions Factorial Moments If g(x) = (x)k , where( x)k = x(x − 1) ··· (x − k + 1), then E(X )k is called the k-th factorial moment.
    [Show full text]