1. Foundations of Numerics from Advanced Mathematics

Transcription

1 Numerical Programming I (for CSE), Hans-Joachim Bungartz page 1 of 50

2 The main purpose of this first chapter (about 4 weeks) is to recall those topics from your Advanced Mathematics courses (linear algebra, calculus, stochastics) typical for the first two years of bachelor s programs in science and engineering that are of particular importance for numerical algorithms and, hence, for the whole CSE master s program. We do this, since you can hardly go successfully through a thorough numerical education without these foundations; since we made the experience that the CSE freshmen s mathematical backgrounds are quite heterogeneous (and not always at hand...); since TUM s CSE program has a methodological (i. e. mathematical and informatical) point of view that goes beyond the usual and widespread engineering approach and way of thinking; and since the two numerics courses have been the most serious roadblock for CSE students since the program s launch (too high failure rates something we want to reduce without touching the level of education). If you are familiar with all this stuff, don t feel bored just consider this as a warm-up to the numerical contents to be discussed later on in this course. Numerical Programming I (for CSE), Hans-Joachim Bungartz page 2 of 50

3 We also changed the name of the courses from Numerical Analysis to Numerical Programming, to indicate that there are mathematical topics to be addressed, but with a clear focus on algorithmics, programming, and applications (instead of proofs etc.). This introductory part won t be a complete lecture with all explanations etc. Rather, it will be a guided tour through important topics, mentioning notions and buzzwords that should have some meaning for you. If they don t, you know that you have to close the gaps as soon as possible, with the help of the references provided or by doing additional exercises etc. Also use the tutorials to refresh your knowledge! Numerical Programming I (for CSE), Hans-Joachim Bungartz page 3 of 50

4 1.1. Mathematical Essentials and Notation Symbols and Notions Everyone familiar with the symbol ; the symbols, 1, and (so-called quantifiers); nx the symbols and Y i=1 i k; the notions maximum, minimum, infimum, and supremum; Kronecker symbol δ ij ; the Landau symbol O(N), O(h 2 ) the symbol ; the meaning of sufficient and necessary; the meaning of iff: sufficient and necessary; the meaning of associative, commutative, and distributive? Numerical Programming I (for CSE), Hans-Joachim Bungartz page 4 of 50

5 Numbers Booleans: true/false; logical operations; relations of logics to set theory (see below) natural numbers, integers N, Z: factorials; binomial coefficients; Pascal s triangle rational numbers Q: countable/non-countable real numbers R: field property (allows for arithmetic operations) order property (allows for comparison) completeness property (each interval nesting defines exactly one real number) supremum/infimum property Q is dense in R different classes of irrational numbers: 2, e,... complex numbers C: imaginary unit i, Re, and Im; conjugate complex fundamental theorem of algebra: each polynomial of degree n with complex coefficients has at least one complex root what else can be said of roots of polynomials? Numerical Programming I (for CSE), Hans-Joachim Bungartz page 5 of 50

6 Sets notions of sets, subsets, and elements set operations: union, intersection, difference, complement symbols,, power set Cartesian product of sets appearances: explicit {1, 2, 3,...} implicit {x R : f(x) = 0} already here a bit of topology: open sets, closed sets, bounded sets, compact sets Numerical Programming I (for CSE), Hans-Joachim Bungartz page 6 of 50

7 Relations definition: relation R between two sets A and B as a subset of A B: R A B notation: arb or (a, b) R important examples for A = B: <,, >,,... properties of relations: transitive reflexive symmetric asymmetric antisymmetric connex notion of an equivalence relation Numerical Programming I (for CSE), Hans-Joachim Bungartz page 7 of 50

8 Mappings and Functions mapping or function (here used in a synonymous way) write x y properties of mappings: injective surjective bijective f 1 (x) =? f : A B : x A 1 y B such that y = f(x); inverse mapping: existence and meaning Numerical Programming I (for CSE), Hans-Joachim Bungartz page 8 of 50

9 Building Blocks of a math course / book / presentation: definition: new notions etc. are defined and, thus, introduced theorem / proposition: a central statement, typically consisting of conditions and a conclusion ( if this and that holds, then the following is valid... ) the more restrictions are made, the more can be concluded (but also the less general the statements are) lemma: similar to a theorem w.r.t. its structure, but usually only an auxiliary statement of minor importance by itself (that marks just a step on the way to a theorem, e.g.) corollary: a statement that follows immediately from a previous theorem etc. proof: a precise argumentation showing clearly that a theorem, lemma, or corollary is correct Note that all this is typically formulated as general and generic as possible a fact which is frequently misinterpreted as not concrete or without practical relevance. Numerical Programming I (for CSE), Hans-Joachim Bungartz page 9 of 50

10 A Short Remark on Proofs Why proofs or how much of proofs? Proofs are the essence of mathematical argumentation they make the latter rigorous. Proofs are a permanent source of misunderstandings and problems: math professors often do not want to do anything without proofs even in courses for non-mathematicians non-math students often think that only the results or statements are relevant, but not the proofs (which they suppose to be something for hardcore mathematicians only) note that both points of view are problematic hence: proofs for non-mathematicians (such as CSE students)? yes, if the way of proving a statement helps to understand it no, if just for itself (i.e. just to prove it) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 10 of 50

11 Standard Proof Techniques forward: A B C D by contradiction ( what if ): D... A by counterexample: to refute the assertion that all students are smart, just find one stupid and the job is done complete search: to prove that all students are smart, check them all mathematical / complete induction: show the statement for n = 1, and show the conclusion from n to n + 1 (does it work for the smart student example?) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 11 of 50

12 1.2. Linear Algebra Mathematical Structures a mathematical structure consists of one or several sets and one or several operations defined on the set(s) special elements: neutral element (of an operation) inverse element (of some element x) a group: a structure to add and subtract a field: a structure to add, subtract, multiply, and divide a vector space: a set with additional properties, allows for addition and multiplication with scalars note: sometimes, the association with classical (geometric) vectors is helpful, sometimes it is more harmful Numerical Programming I (for CSE), Hans-Joachim Bungartz page 12 of 50

13 Vector Spaces a linear combination of vectors linear (in)dependence of a set of vectors the span of a set of vectors a basis of a vector space definition? why do we need a basis? is a vector s basis representation unique? is there only one basis for a vector space? the dimension of a vector space does infinite dimensionality exist? important applications: (analytic) geometry numerical and functional analysis: function spaces are vector spaces (frequently named after mathematicians: Banach spaces, Hilbert spaces, Sobolev spaces,...) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 13 of 50

14 Linear Mappings definition in the vector space context; notion of a homomorphism image and kernel of a homomorphism matrices, transposed and Hermitian of a matrix relations of matrices and homomorphisms meaning of injective, surjective, and bijective for a matrix; rank of a matrix meaning of the matrix columns for the underlying mapping matrices and systems of linear equations basis transformation and coordinate transformation mono-, epi, iso-, endo-, and automorphisms Numerical Programming I (for CSE), Hans-Joachim Bungartz page 14 of 50

15 Determinants definition properties meaning occurrences Cramer s rule Numerical Programming I (for CSE), Hans-Joachim Bungartz page 15 of 50

16 Eigenvalues notions of eigenvalue, eigenvector, and spectrum similar matrices A, B: S : B = SAS 1 (i.e.: A and B as two basis representations of the same endomorphism) resulting objective: look for the best / cheapest representation (diagonal form) important: matrix A is diagonalizable iff there is a basis consisting of eigenvectors only characteristic polynomial, its roots are the eigenvalues Jordan normal form important: spectrum characterizes a matrix many situations / applications where eigenvalues are crucial Numerical Programming I (for CSE), Hans-Joachim Bungartz page 16 of 50

17 Scalar Products and Vector Norms notions of a linear form and a bilinear form scalar product: a positive-definite symmetric bilinear form examples of vector spaces and scalar products vector norms: definition: positivity, homogeneity, triangle inequality meaning of triangle inequality examples: Euclidean, maximum, and sum norm normed vector spaces Cauchy-Schwarz inequality notions of orthogonality and orthonormality turning a basis into an orthonormal one: Gram-Schmidt orthogonalization Numerical Programming I (for CSE), Hans-Joachim Bungartz page 17 of 50

18 Matrix Norms definition: properties corresponding to those of vector norms plus sub-multiplicativity: AB A B plus consistency Ax A x matrix norms can be induced from corresponding vector norms: Euclidean, maximum, sum A := max Ax x =1 alternative: completely new definition, for example Frobenius norm (consider matrix as a vector, then take Euclidean norm) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 18 of 50

19 Classes of Matrices symmetric: A = A T skew-symmetric: A = A T Hermitian: A = A H = ĀT s.p.d. (symmetric positive definite): x T Ax > 0 x 0 orthogonal: A 1 = A T (the whole spectrum has modulus 1) unitary: A 1 = A H (the whole spectrum has modulus 1) normal: AA T = A T A or AA H = A H A, resp. (for those and only those matrices there exists an orthonormal basis of eigenvectors) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 19 of 50

20 1.3. Calculus Functions Revisited notions of a function, its range, and its image graph of a function isolines and isosurfaces sums and products of functions composition of functions inverse of a function: when existing? simple properties: (strictly) monotonous explicit and implicit definition parametrized representations (curves,...) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 20 of 50

21 Continuity remember the ε and the δ! definition of local ( in x 0 ) and global continuity ( x ) what about sums, products, quotients,... of continuous functions? what about compositions of continuous functions? what about continuity of the inverse? intermediate value theorem continuous functions on compact sets maximum and minimum value uniform continuity Numerical Programming I (for CSE), Hans-Joachim Bungartz page 21 of 50

22 Limits meaning of ε 0 and N and x accumulation point of a set limit (value) of a set limits from the left or from the right, respectively: f(x+), f(x ) limits at infinity: lim x f(x) infinite limits: f(x) how can discontinuities look like? jumps: f(x+) f(x ) holes: f(x+) = f(x ) f(x) second kind: f(x) = 0 in x = 0 and f(x) = sin ` 1 x elsewhere Numerical Programming I (for CSE), Hans-Joachim Bungartz page 22 of 50

23 Sequences definition of a sequence: a function f defined on N if f(n) = a n, write (a n) or a 1, a 2, a 3,... bounded / monotonously increasing / monotonously decreasing sequences notion of convergence of a sequence: existence of a limit for n Cauchy sequence subsequences Numerical Programming I (for CSE), Hans-Joachim Bungartz page 23 of 50

24 Series notion of an (infinite) series elements of a series partial sums of a series convergence defined by convergence of the sequence of the partial sums convergence and absolute convergence examples: geometric series: P k=1 xk = 1 1 x harmonic series: P k=1 1 k = alternating harmonic series: P k=1 ( 1)k 1 1 k = ln(2) criteria for convergence: quotient and root criterion power series: P k=0 a k(z a) k coefficients a k and centre point a radius of convergence R: absolute convergence for z a < R identity theorem for power series re-arrangement sums of series, nested series, products of series (Cauchy product) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 24 of 50

25 Differentiation first step: functions f of one real variable, complex values allowed derivative or differential quotient of f: defined via limit process of difference quotients write f or f or df dx geometric meaning? local and global differentiability derivative from the left / from the right rules for the daily work: derivative of f + g, fg, and f/g? derivative of f(g) (chain rule)? derivative of the inverse function? higher derivatives f (k) (x); meaning notion of continuous differentiability smoothness of a function space of k-times continuously differentiable functions: C k Numerical Programming I (for CSE), Hans-Joachim Bungartz page 25 of 50

26 Differential Calculus of one Real Variable notion of a global/local minimum/maximum local extrema and the first derivative mean value theorem: ξ (a, b) : f (ξ) = f(b) f(a) b a monotonous behaviour and the first derivative local extrema and the second derivative rule of de l Hospital notions of convexity and concavity convexity/concavity and the second derivative notion of a turning point turning points and the second derivative Numerical Programming I (for CSE), Hans-Joachim Bungartz page 26 of 50

27 Function Classes (1) polynomials definition, degree, sums and products, division with rest, identity theorem, roots and their multiplicity rational functions poles and their multiplicity, partial fraction decomposition exponential function and logarithm characterising law of the exponential function: exp(s + t) = exp(s) exp(t) or y = y (functional equation of natural growth) series expansion of the exponential function, speed of growth natural logarithm as exp s inverse: y = exp(x) = e x, x = ln(y) functional equation: ln(xy) = ln(x) + ln(y) exponential function and logarithm for general basis a: a x := e x ln a, log a (y) := ln y ln a Numerical Programming I (for CSE), Hans-Joachim Bungartz page 27 of 50

28 Function Classes (2) hyperbolic functions cosh(z), sinh(z),... trigonometric functions sin(x), cos(x): solutions of y (2) + y = 0 geometric meaning? Euler s formula: e ix = cos(x) + i sin(x) derivatives, addition theorem periodicity series expansion Numerical Programming I (for CSE), Hans-Joachim Bungartz page 28 of 50

29 Integral Calculus of one Variable Riemann integral, upper and lower sums approximation by staircase functions properties: linearity monotonicity mean value theorem: Z b ξ (a, b) : f(x)dx = (b a) f(ξ) a main theorem of differential and integral calculus: define F (x) := R x a f(t)dt then R b a f(t)dt = F (b) F (a) rules for everyday work: partial integration: Z Z uv dx = uv vu dx substitution: Z b f(t(x))t (x)dx = a Z t(b) f(t)dt t(a) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 29 of 50

30 Local Approximation: Taylor Polynomials and Series local approximation of functions with polynomials generalization of the tangent approximation used for the definition of the derivative Taylor polynomials: let f be n-times differentiable in a we look for a polynomial T with T (k) = f (k) for k = 0, 1,..., n obviously: nx 1 T (x) := k! f (k) (a)(x a) k k=0 unique, degree n, write T nf(x; a) remainder R n+1 (x) := f(x) T nf(x : a) Taylor series: R n+1 (x) = f (n+1) (ξ) (x a)n+1 (n + 1)! for infinitely differentiable functions (exp, sin, cos,...) sum up to instead of n only examples Numerical Programming I (for CSE), Hans-Joachim Bungartz page 30 of 50

31 Global Approximation: Uniform Convergence convergence of sequences of functions f n defined on D: pointwise: for each x D; then defines a function f(x) := lim fn(x) n problems: are properties such as continuity or differentiability inherited from the f n to f, and how to calculate derivatives or integrals of f? i.e., can the order of limit processes be changed? therefore the notion of uniform convergence: definition: f n f D 0 for n with that, the inheritance and change-order problems from above are solved! criteria: Cauchy,... approximation theorem of Weierstrass: each continuous function f on a compact set can be arbitrarily well approximated with some polynomial Numerical Programming I (for CSE), Hans-Joachim Bungartz page 31 of 50

32 Simple Differential Equations notion of a differential equation ordinary: one variable partial: more than one variable (several spatial dimensions or space and time) examples: growth: ẏ = k y or ẏ = k(t, y) y oscillation: ÿ + y = 0 or similar example of an analytic solution strategy: separation of variables y = g(x) h(y), y(x) = y 0 formal separation: integration of the left and right side some requirements for applicability dy h(y) = g(x)dx Numerical Programming I (for CSE), Hans-Joachim Bungartz page 32 of 50

33 Periodic Functions target now: periodic functions, period typically 2π trigonometric polynomials definition: T (x) := nx c k e ikx = a X n (a k cos(kx) + b k sin(kx)) k= n k=1 (coefficients c k, a k, and b k are unique) formula for the coefficients: c k = 1 2π T is real iff all a k, b k are real iff c k = c k Z 2π T (x)e ikx dx 0 Weierstrass: 2π-periodic continuous functions can be arbitrarily well approximated by trigonometric polynomials Numerical Programming I (for CSE), Hans-Joachim Bungartz page 33 of 50

34 Fourier Series consider vector space of 2π-periodic complex functions f on R Z 1 2π Fourier coefficients: ˆf(k) := f(x)e ikx dx 2π 0 nx Fourier polynomial: S nf(x) := ˆf(k)e ikx Fourier series: sine-cosine representation of S nf: coefficients: k= n X ˆf(k)e ikx S nf(x) = a 0 n 2 + X (a k cos(kx) + b k sin(kx)) k=1 a k = ˆf(k) + ˆf( k) = 1 Z π f(x) cos(kx)dx π π b k = i( ˆf(k) ˆf( k)) = 1 Z π f(x) sin(kx)dx π π all a k vanish for odd f, all b k vanish for even f Numerical Programming I (for CSE), Hans-Joachim Bungartz page 34 of 50

35 Functions of Several Variables f now defined on R n or a subset of it notion of differentiability: now via existence of a linear map, the differential directional derivatives partial derivatives prominent differentiability criterion: existence and continuity of all partial derivatives the gradient of a scalar function f and its interpretation the Jacobian of a vector-valued function f mean value theorem higher partial derivatives, Taylor approximation, Hessian local minima and maxima, criteria saddle points Numerical Programming I (for CSE), Hans-Joachim Bungartz page 35 of 50

36 Integration over Domains a huge field, from which we only mention a few results theorem of Fubini: shows that, in many cases, a multi-dimensional integration domain can be tackled dimension by dimension statement (we neglect the requirements, for which more integration theory is needed): Z Z Z f(x, y)d(x, y) = X Y Y f(x, y)dx X related to Cavalieri s principle will also be of relevance for numerical quadrature transformation theorem: a generalisation of integration by substitution statement, again without requirements: Z f(t (x)) det T (x) dx = U «Z Z dy = X Y Z f(y)dy V allows for a change of the coordinate system (polar coordinates), e.g. «f(x, y)dy dx Numerical Programming I (for CSE), Hans-Joachim Bungartz page 36 of 50

37 Gauss Theorem we further generalise integration, now allowing for integration over hyper-surfaces (a sphere, e.g.) this is important for the physical modelling in many scenarios (heat flux through a pot s surface,...) the famous Gauss theorem allows to combine integrals over volumes and surfaces, which occurs in the derivation of many physical models (conservation laws) and, hence, is of special relevance for CSE prerequisites: a vector field: a vector-valued function on R n (example: the velocity field in fluid mechanics) the divergence of a vector field F : nx div F (x) = i F i (x) finally the Gauss theorem: several regularity assumptions needed Z div F dx = G i=1 Z G F ds Numerical Programming I (for CSE), Hans-Joachim Bungartz page 37 of 50

38 1.4. Stochastics and Statistics Not a primary ingredient of CSE, but you should have at hand at least some basic knowledge (for things to come later: Monte-Carlo-based simulations, stochastic PDE,...): probability theory: subfield of mathematics providing the apparatus to formalise random events and studying what can be derived within that formalism; typical question: if the world looks this and that, what can we say concerning the results of a random experiment? mathematical statistics: not that firmly connected to mathematics; descriptive statistics deals with the presentation of huge data sets, inductive statistics deals with conclusions from measured or observed values of random variables to their underlying properties stochastics: sometimes used in a synonymous way to probability theory, sometimes as a notion covering probability theory and statistics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 38 of 50

39 Combinatorics Combinatorics deals with counting possible cases of a certain characteristics in a discrete and finite universe. The most famous and important example: Select k n elements from a set of n elements. Remember the four cases: with repetition, ordered: n k without repetition, ordered: k 1 Y n k = without repetition, unordered: i=0 (n i) = n n! = k (n k)!k! n! (n k)! with repetition, unordered: n + k 1 k Numerical Programming I (for CSE), Hans-Joachim Bungartz page 39 of 50

40 Discrete Probability Spaces: Events (1) random experiments have results ω i from a discrete (possibly infinite) result set Ω = {ω 1,...} events as subsets of this result set: A, B Ω A B, A B, Ā, A \ B elementary events {ω i } with probabilities p i rules: i : p i [0, 1], P Ω p i = 1 handling probabilities: p(ā) = 1 p(a), p(a B) = p(a)+p(b) p(a B), A B p(a) p(b),... conditional probabilities (check that p(. B) are probabilities): p(a B) := p(a B) p(b) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 40 of 50

41 Discrete Probability Spaces: Events (2) multiplication theorem: p(a 1... A n) = p(a 1 ) p(a 2 A 1 ) p(a 3 A 1 A 2 )... p(a n A 1... A n 1 ) theorem of total probability for pairwise disjoint A i : p(b) = X i p(b A i ) p(a i ) theorem of Bayes: p(a j B) = p(a j B) p(b) = p(b A j) p(a j ) P i p(b A i) p(a i ) Numerical Programming I (for CSE), Hans-Joachim Bungartz page 41 of 50

42 Independence of Events intuitive definition: two events A, B are independent, if there is no influence among each other: p(a B) = p(a), p(b A) = p(b) formal definition: independence, if p(a B) = p(a) p(b) examples? independence of a set of events A 1,..., A n: 0 1 \ A i A = Y p(a i ) I {1,..., n} i I i I Numerical Programming I (for CSE), Hans-Joachim Bungartz page 42 of 50

43 Discrete Random Variables definition: X : Ω R X defines events: X = x := {ω i Ω : X(ω i ) = x},... discrete density (function): f X (x) := p(x = x) discrete distribution (function): F X (x) := p(x x) expectation or mean value: E(X) := X x x p(x = x) variance: V (X) := E `(X E(X)) 2 rules how to work with E(X) and V (X): E(aX + b) = ae(x) + b, V (ax + b) = a 2 V (X),... Numerical Programming I (for CSE), Hans-Joachim Bungartz page 43 of 50

44 Important Discrete Distributions Bernoulli distribution: binary random variable results 1 and 0 with probabilities p and 1 p, resp. Binomial distribution sum of n Bernoulli variables p(x = k) = `n pk (1 p) k n k geometric distribution repeat a Bernoulli experiment until success (result 1) p(x = i) = p(1 p) i 1 Poisson distribution counts events possible results 0, 1, 2, 3,... density f X (i) = e λ λ i i! exercise: think about examples, calculate E(X) and V (X) for all Numerical Programming I (for CSE), Hans-Joachim Bungartz page 44 of 50

45 Continuous Probability Spaces motivation: think about the differences of discrete and continuous realities and probability spaces sums get integrals now: expectation value: density: distribution: p(a) := properties of f X and F X? E(X) := Z f X (t)dt, A F X (x) := Z tf X (t)dt Z f X (t)dt = 1 Z x f(t)dt Numerical Programming I (for CSE), Hans-Joachim Bungartz page 45 of 50

46 Important Continuous Distributions uniform distribution no differences in probability density: f X (x) = normal or Gaussian distribution j 1 b a for x [a, b], 0 otherwise. frequent in reality, especially in limit cases, see slide on asymptotics density: f X (x) = 1 2πσ e (x µ)2 2σ 2, F X (x) =: Φ(x) N (µ, σ) negative exponential distribution describes time between events density: j λ e λx if x 0, f X (x) = 0 otherwise exercise: think about examples and calculate E(X) and V (X) for all Numerical Programming I (for CSE), Hans-Joachim Bungartz page 46 of 50

47 Asymptotics (1) inequality of Markov: inequality of Chebyshev: p(x t) E(X) t. theorem of large numbers: p( X E(X) t) V (X) t 2. given random variables X i, i = 1, 2,..., iid (independent and identically distributed) with mean value µ and variance σ 2 define arithmetic average Z n of X 1,..., X n: Z n := 1 n nx X i. i=1 given ε, δ > 0 and n N with n σ2 εδ 2 then it holds p( Z n µ δ) ε Numerical Programming I (for CSE), Hans-Joachim Bungartz page 47 of 50

48 Asymptotics (2) central limit theorem: given random variables X i, i = 1, 2,..., iid with mean value µ and variance σ 2 > 0 define Y n, n = 1, 2,... as n-th partial sum of X i : Y n := nx X i i=1 let Z n be the normalized random variable of Y n defined as Z n := Yn nµ σ n Then: All Z n are asymptotically standardized normally distributed, i. e. lim Zn N (0, 1) n Particularly, it holds for the series of distribution functions F Zn of Z n lim F Z n n (x) = Φ(x) x R Numerical Programming I (for CSE), Hans-Joachim Bungartz page 48 of 50

49 Inductive Statistics (1) (random) sample: repeat a random experiment record results as an iid sequence (ω i ) or (x i ), resp. objective: learn about reality with the help of the sample first technique: estimators: Y := g(x 1,..., X n) properties: unbiased, variance-reducing, efficient, consistent examples: Y := X := 1 nx X i for E(X) n i=1 Y := S 2 1 nx := (X i X) 2 for V (X) n 1 i=1 maximum likelihood principle and estimators Numerical Programming I (for CSE), Hans-Joachim Bungartz page 49 of 50

50 Inductive Statistics (2) second technique: confidence intervals construct two estimators U 1, U 2 to define an interval for an unknown parameter θ (E(X), V (X),...): third technique: tests p(u 1 θ U 2 ) 1 α hypothesis H 0 and alternative H 1 on the size of an unknown parameter E(X) = 27.2, p < 0.9,... make a sample determine a range where to reject the hypothesis error types: error of first kind, error of second kind maximum risk of first kind: significance level of the test exercise: study simple examples Numerical Programming I (for CSE), Hans-Joachim Bungartz page 50 of 50