) k ( 1 λ ) n k. ) n n e. k!(n k)! n
|
|
- Ezra Goodwin
- 5 years ago
- Views:
Transcription
1
2 k ) λ ) k λ ) λk k! e λ ) π/!. e α + α) /α e k ) λ ) k λ )! λ k! k)! ) λ k λ k! λk e λ k! λk e λ. k! ) k λ ) k k + k k k ) k ) k e k λ e k ) k EX EX V arx) X Nα, σ ) Bp) Eα) Πλ) U, θ) X Nα, σ ) E ) X α σ E X α ) σ EX α V arx) σ EX EX) + V arx) α + σ X Bp) EX P X ) p EX P X ) p V arx) EX EX) p p X Eα) EX xαe αx dx yαx α ye y dy α Γ) α, EX x αe αx dx α y e y dy α Γ3) α V arx) EX EX) α X Πλ) EX λ k k k )! e λ λ k λk k! e λ λ EX k kλ k k )! e λ k k )λ k k )! e λ + V arx) EX EX) λ + λ λ λ X U, θ) EX θ x θ dx θ EX θ k x θ λ k k )! e λ λ k λ k k! e λ + EX λ + λ, θ θ dx 3 V arx) 3 θ 4 θ θ U[, θ] fx θ) θ x θ, < θ <
3 m θ) E θ X θ m θ) E θ X θ 3 θ m X) X, θ m X ) 3 X. θ θ V ar θ X) m θ )) ) θ. θ E θ X 4 X θ E θ e tx e tx tθ dx etθ ) k θ tθ k + )!. E θ e tx k tk k! E θ X k E θ X k θk k+ E θ X 4 θ4 5 V ar θ X 4 θ ) m θ )) θ > 5 θ θ 45 7 k 9 θ 3 4. λ Πλ) φλ) i d d φλ) λ + λ dλ dλ px i λ) λ i X i i X i! e λ. ) i X i X i! + X i. λ i λ < i X i φx) λ > i X i φx) φλ) λ i X i λ Πλ) λ i X i α Eα) φα) px i α) [, ) X i ) α e α i Xi [, ) Y ), i Y X,, X } φα) φα) i i d d φα) dα dα α α X i ) α d dα φα) α <. φα) α α / X i Xi i X i X α Eα)
4 e θ x x θ fx θ) x < θ, < θ < θ φθ) e θ i Xi [θ, )Y ) e θ i X i θ Y θ > Y, Y X,, X } e θ i X i θ φθ) θ Y θ θ X,, X } fx θ) e x θ < x <, < θ < θ φθ) e i Xi θ. m X d E X m ) E X d ). d X x x i x i m i x i d, m x,, x } d d x,, x } φθ) θ X X θ X X Iλ) Πλ) fx λ) λx x! e λ lx λ) λ+x λ x! l x λ) x λ Iλ) E λ l X λ) E λx λ λ Iα) Nα, σ ) σ α x α) lx α) e σ πσ Iα) σ x α) σ πσ) l x α) σ Bp) p
5 fx p) p x p) x e x p p p) ap)bx)e cp)dx), ap) p bx) cp) p p dx) x S i dx i) i X i X E p S p X p Nα, σ ) σ σ α α fx α) πσ e x σ e α σ e αx σ fx α) aα) e α x σ bx) e σ πσ cα) α σ dx) x S i dx i) i X i X E α S α X α p p p X X p ξp) Bα, β) E[p] α α+β.5 E[p X,, X ].9 α β p ξx X x,, X x ) P p x X x,, X x ) P X x,, X x p x)p p x) P X x,, X x p y)p p y)dy Cx i x i x) i x i) x α x) β Cx i xi+α x) β+ i xi) C ξp x X x,, X x ) x Γ + α + β) C Γ i x i + α)γ i x i) + β), α 7 E[p X,, X ] β 9 7 xp p x X X )dx Γ + α + β) x Γα)Γ + β) xα x) +β dx α + α + β +. α α+β.5 α+ α+β+.9,
6 p p ξ.).7 ξ.).3. p ξ. X x,, X x ) P p. X x,, X x ) P X x,, X x p.)p p.) P X x,, X x p.)p p.) + P X x,, X x p.)p p.) ξ. X x,, X x ) θ θ θ Γα, β) E[θ] α β V arθ) α β α β α β 5, α β Eα) α α α ξx) α Γa, b) E[α] a b. V arα) a b a 5 b 5 ξx X x,, X x ) P X x,, X x α x)p α x) P X x,, X x α y)p α y)dy x e x i x i b a Γa) xa e bx Costat b + i x i) +a x +a e b+ i xi)x Γ + a) +a b + X) x +a e b+ X)x. Γ + a) X 3.8 a 5 b 5 α p p α 5 β p
7 p Γα + β + ) ξx X,, X ) Γα + i X i)γβ + i X i) xα+ i Xi) x) β+ i X i). p p E[p X,, X ] x Γ35)x5 x) 8 dx 6 Γ6)Γ9) 35. Πλ) λ λ α 3 β λ λ ξx X,, X ) λ E[λ X,, X ] β + )α+ i Xi x α+ Γα + i X i) i Xi e β+)x x 3+3 x 3+3 e x dx 8. Γ3 + 3). Γα, β) β α Γα, β) β [ ] β α fx,, x α) x x ) α e β i x i x,,x Γα) } ux,, x )vt x,, x ), α), ux,, x ) e β [ ] i x i x,,x } vt, α) β α Γα) T α T x,, x ) i x i T X,, X ) i X i fx θ) aθ)bx)e cθ)dx) X X fx,, x θ) aθ) i bx i )e cθ) i dxi) ux,, x )vt x,, x ), θ), T x,, x ) i dx i) vt, θ) aθ) e tcθ) ux,, x ) i bx i) T X,, X ) i dx i)
8 Γα, β) β α fx,, x α, β) [ ] β α x x ) α e β i xi x,,x Γα) } ux,, x )vt x,, x ), T x,, x ), α, β), [ ] ux,, x ) x,,x } vt, t, α, β) β α Γα) t α e βt T x,, x ) i x i T x,, x ) i x i i X i, i X i) X X Eα) α α Eα) fx,, x α) α e α i xi x,,x }. x,, x } < fx,, x α) x,, x } α α fx,, x α) α α α i x x. i T x,, x ) x ux,, x ) x,,x } vt, α) α e α t fx,, x α) ux,, x )vt x,, x ), α) α α X α X X U[, θ] Y X,, X }. θ cy E θ cy θ). θ cy Y t t θ} E θ cy θ) θ ct θ) t dt + c ) + c + θ +. 86, 8, 76, 49, 84, 9, 58, 39, 75, 48, 5,, 4, 53, 9, 57, 3, 49, 35, 3 µ σ
9 X X X X) c c c P χ 9 c ) P χ 9 c ) α.5 P t 9 c) P t 9 c) α.5 c c 3.4 c.79 σ [ X X) ), X X) ] ) [33.8, 43.] c c µ [ ] X c X X) ), X + c X X) ) [48., 65.6]. f x) f x) x f x) f x) x x X fx) f x) f x) H : fx) f x), H : fx) f x). αδ) + βδ) αδ) + βδ) αδ) βδ) αδ) P δ H ) βδ) P δ H ) ξ) 3 ξ) 3 ξ)p δ H ) + ξ)p δ H ) f X) f ξ) X) X ξ) H : X < 4 δ H : X > 4 H H : X 4. δ H : X δ 4 H : X > 4. αδ ) + βδ ) P δ H ) + P δ H ) P X > 4 ) + P X 4 ) 4 4 dx + xdx 7 8.
10 αδ). βδ) βδ) c ) f X) P f X) < c.. c 5 9 ) ) f X) P f X) < c P X < c P X > c ) c } c ).. δ H : X 5 9 H : X < 5 9 αδ). βδ) βδ) P δ H ) P X 9.8. H H H P p. H P p.4 X X α.5 fx,, x p) p i x i p) i x i) f X, X ). X+X.8 X+X) f X, X ).4 X+X.6 X+X) f X, X ) f X, X ).5X +X ) X +X 4 ). 3 c P f X,X ) f X,X ) < c ) α.5 ) f X, X ) P f X, X ) < c 6 9 <c} <c} <c}.3. c p ) ) f X, X ) P f X, X ) < c f X, X ) + p)p f X, X ) c.5. ) P f X,X ) f X,X ) < c ) P f X,X ) f X,X ) c c 3.5 c 4, 3 ].4 + p)p X, X X, X ).5,
11 p 3 3 α.5 H :.5 X+X 4 X+X ) 3) > 3 δ H :.5 X+X 4 X+X ) 3) < 3 H H :.5 X+X 4 X+X ) 3) H p 3 3 H p 3 ) ) f X, X ) P δ H ) P f X, X ) < c f X, X ) + p)p f X, X ) c P X X ) + 3 P X, X X, X ) , X X µ σ H : µ H : µ δ α δ).5 } f X,, X ) f X,, X ) i [X i µ ) X i µ ) ] σ ) f X,, X P f X,, X ) < c P i X i < c) i e X i. ) i P X i ) c < σ σ ) c Φ. ) Φ c.5 c.88 δ i H : e X i.88 δ i H : e X i <.88. ) i P δ H ) P e X i < c ) c + Φ.94.
12 α.5 Aa /4 AA / Aa /4 aa Aa AA Aa aa χ p P AA) p P Aa) p 3 P aa) H : p p, p p, p 3 p 3 H :, p, p p 3 4 v i X i AA} v i X i Aa} v 3 i X i aa} T 3 k v k p k ) p k χ δ H : T c H : T > c, c.5 α χ 3 c, ) c 5.99 T
13 χ x.84 x.53 x 3.53 x 4.84 Φx k ).k k,, 3, 4) B,.84] B.84,.53] B 3.53,.53] B 4.53,.84] B 5.84, ) p k P X B k ) k,, 3, 4, 5) H : p p p 3 p 4 p 5. H :. v k i X i B k } k,, 3, 4, 5) v 5 v v 3 7 v 4 v v k 5.) T 5. k 5.4. χ δ H : T c H : T > c, c.5 α χ 5 c, ) c H N N 6 θ θ H N, N 36, N 3 4, N 4 36, N 5, N 6 4. Q H fθ, θ ) θ N θ N θ θ ) N 3 θ θ ) N 4 [θ θ θ )] N 5 [θ θ θ )] N 6. A <
14 fθ, θ ) N θ + N θ + N 3 θ θ ) + N 4 + θ + θ ) +N 5 [ + θ + θ θ )] + N 6 [ + θ + θ θ )]. fθ, θ ) θ θ N3+N5+N6 θ θ θ fθ, θ ) N+N4+N5 θ θ fθ, θ ) N +N 4 +N 6 θ N 3+N 5 +N 6 θ θ. θ θ N i θ θ θ. θ.5 T N 5. ) 5. + N 5.5 ) N ) N 4 5.) + N 5 5.) + N 6 5.3) T χ 6 χ 3 c 7.85 χ 3c, ).5 T c T 4.37 < c θ < θ < ) p θ) 4, p 4θ θ) 3, p 6θ θ), p 3 4θ 3 θ), p 4 θ 4. H : p θ) 4, p 4θ θ) 3, p 6θ θ), p 3 4θ 3 θ), p 4 θ 4, θ, ) H :. fθ) 4 i i pn i i N i fθ) 4N θ) + N [ 4 + θ + 3 θ)] + N [ 6 + θ + θ)] +N 3 [ θ + θ)] + 4N 4 θ.
15 θ θ θ.4 p θ).96 p θ).3456 p θ).3456 p 3 θ).536 p 4 θ).56 T χ 33.96) ) ) 9.56) ).3456 T χ 5 χ 3 α c χ 3c, ) α H : T c δ H : T > c α c 7.85 χ 3c, ).5 T > c N 7 N 6 N 3 5 N 3 N 3 N 3 9 N + 7 N + 5 N + 47 N + 9 N +3 4 χ T [7 7 47/478) /7 47) /478) /7 9) /478) /7 4) /478) /5 47) /478) /5 9) /478) /5 4)] α : T c δ : T > c, c χ )3 )c, ) α δ α c 5.99 χ c, ).5 T > 5.99 T
16 O A B AB Rh O A B AB Rh Rh N 8 N 89 N 3 54 N 4 9 N 3 N 7 N 3 7 N 4 9 N + 44 N + 56 N + 95 N + 6 N +3 6 N +4 8 χ T 8.6. [ /3) /44 95) /3) /44 6) /3) /44 6) /3) /44 8) /3) /56 95) /3) /56 6) /3) /56 6) /3) /56 8)] 3 α : T c δ : T > c, c χ )4 )c, ) α δ α c 7.85 χ 3c, ).5 T >
17 N 57 N 4 N 3 5 N 87 N N 3 4 N 3 3 N 3 6 N N + 49 N + 5 N N + 47 N + 9 N +3 4 T α : T c δ : T > c, c χ 3 )3 )c, ) α δ α c χ 4c, ).5 T > [, ] H : H : F x) F x) x. H : D.35 δ H : D >.35 D x F x) x 5 D A
18 x.4 F x) F x) D H fx) 3 < x fx) < x <. fx) x 3 F x) x < x x+ < x < x. H : H : F x) F x). H : D.35 δ H : D >.35 D
19 x.66 F x) F x) D H / [, ] / θ θ [, ] θ θ P θ ) P θ ).5 P θ X x,, X 5 x 5 ) P X x,, X 5 x 5 θ )P θ ) P X x,, X 5 x 5 θ )P θ ) + P X x,, X 5 x 5 θ )P θ ) P X x,, X 5 x 5 θ ) P X x,, X 5 x 5 θ ) + P X x,, X 5 x 5 θ ) i <x i.5} i.5<x i<} β β σ y x.5 β i x i y i i x i y i β β σ β XY XȲ X X) , β Ȳ β X.47, σ Y i β β X i ).45. i α. α α χ, c ) α.5 χ c, ) α.5 c.736 c σ [ σ c, σ c ] [.9,.65] β c t c, c) α.9 c.8595 β [ ] β c [.88,.5876]. σ ) X X) ), β + c σ ) X X) )
20 β [ σ β c + X) ) X X), β + c σ + X) )] [.64,.38]. X X) x.5 ŷ y β + β.5.8 y x.5 [ σ Ŷ c + + X ) X) X X), Ŷ + c σ + + X )] X) [.386,.94]. X X) H : β H : β <, β N ) X X) β β ) σ σ σ β X X) ) T ) β β σ σ σ ) X X) ) ) σ β, X X) ) χ β β t β T T H : T > c δ H : T c, c t 8, c). c.3968 T 5.3 > c H σ )
Recursive Route to Mixed Poisson Distributions using Integration by Parts
ISSN 2224-584 (Paper) ISSN 2225-522 (Online) Vol.4, No.4, 24 Recursive Route to Mixed Poisson Distributions using Integration by Parts Abstract Rachel Sarguta * Joseph A. M. Ottieno School of Mathematics,
More informationB.Sc. MATHEMATICS I YEAR
B.Sc. MATHEMATICS I YEAR DJMB : ALGEBRA AND SEQUENCES AND SERIES SYLLABUS Unit I: Theory of equation: Every equation f(x) = 0 of n th degree has n roots, Symmetric functions of the roots in terms of the
More informationSolutions 2017 AB Exam
1. Solve for x : x 2 = 4 x. Solutions 2017 AB Exam Texas A&M High School Math Contest October 21, 2017 ANSWER: x = 3 Solution: x 2 = 4 x x 2 = 16 8x + x 2 x 2 9x + 18 = 0 (x 6)(x 3) = 0 x = 6, 3 but x
More informationExercises for Chap. 2
1 Exercises for Chap..1 For the Riemann-Liouville fractional integral of the first kind of order α, 1,(a,x) f, evaluate the fractional integral if (i): f(t)= tν, (ii): f(t)= (t c) ν for some constant c,
More informationLecture 17: The Exponential and Some Related Distributions
Lecture 7: The Exponential and Some Related Distributions. Definition Definition: A continuous random variable X is said to have the exponential distribution with parameter if the density of X is e x if
More informationDirection: This test is worth 250 points and each problem worth points. DO ANY SIX
Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes
More informationStat410 Probability and Statistics II (F16)
Stat4 Probability and Statistics II (F6 Exponential, Poisson and Gamma Suppose on average every /λ hours, a Stochastic train arrives at the Random station. Further we assume the waiting time between two
More informationPractice Examination # 3
Practice Examination # 3 Sta 23: Probability December 13, 212 This is a closed-book exam so do not refer to your notes, the text, or any other books (please put them on the floor). You may use a single
More informationReview For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.
Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationContinuous Random Variables
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables
More informationarxiv: v1 [math.ho] 4 Jul 2007
arxiv:0707.0699v [math.ho] 4 Jul 2007 A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots Leonhard
More informationP (x). all other X j =x j. If X is a continuous random vector (see p.172), then the marginal distributions of X i are: f(x)dx 1 dx n
JOINT DENSITIES - RANDOM VECTORS - REVIEW Joint densities describe probability distributions of a random vector X: an n-dimensional vector of random variables, ie, X = (X 1,, X n ), where all X is are
More informationLIST OF FORMULAS FOR STK1100 AND STK1110
LIST OF FORMULAS FOR STK1100 AND STK1110 (Version of 11. November 2015) 1. Probability Let A, B, A 1, A 2,..., B 1, B 2,... be events, that is, subsets of a sample space Ω. a) Axioms: A probability function
More informationMore on Bayes and conjugate forms
More on Bayes and conjugate forms Saad Mneimneh A cool function, Γ(x) (Gamma) The Gamma function is defined as follows: Γ(x) = t x e t dt For x >, if we integrate by parts ( udv = uv vdu), we have: Γ(x)
More informationOur task The optimization principle Available choices Preferences and utility functions Describing the representative consumer preferences
X Y U U(x, y), x X y Y (x, y ) U(x, y ) (x, y ) (x, y ) (x, y ) (x, y ) (x, y ) U(x, y ) > U(x, y ) (x, y ) (x, y ) U(x, y ) < U(x, y ) U(x, y ) = U(x, y ) X Y U x y U x > x 0 U(x, y) U(x 0, y), y y >
More informationRandom Variables. Random variables. A numerically valued map X of an outcome ω from a sample space Ω to the real line R
In probabilistic models, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. As a function or a map, it maps from an element (or an outcome) of a sample
More informationSolutions for MAS277 Problems
Solutions for MAS77 Problems Solutions for Chapter problems: Inner product spaces 1. No. The only part that fails, however, is the condition that f, f should mean that f. If f is a non-zero function that
More informationSolutions to Dynamical Systems 2010 exam. Each question is worth 25 marks.
Solutions to Dynamical Systems exam Each question is worth marks [Unseen] Consider the following st order differential equation: dy dt Xy yy 4 a Find and classify all the fixed points of Hence draw the
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More informationConditional distributions. Conditional expectation and conditional variance with respect to a variable.
Conditional distributions Conditional expectation and conditional variance with respect to a variable Probability Theory and Stochastic Processes, summer semester 07/08 80408 Conditional distributions
More informationMathematical Statistics
Mathematical Statistics Chapter Three. Point Estimation 3.4 Uniformly Minimum Variance Unbiased Estimator(UMVUE) Criteria for Best Estimators MSE Criterion Let F = {p(x; θ) : θ Θ} be a parametric distribution
More informationActuarial Science Exam 1/P
Actuarial Science Exam /P Ville A. Satopää December 5, 2009 Contents Review of Algebra and Calculus 2 2 Basic Probability Concepts 3 3 Conditional Probability and Independence 4 4 Combinatorial Principles,
More information2017 Financial Mathematics Orientation - Statistics
2017 Financial Mathematics Orientation - Statistics Written by Long Wang Edited by Joshua Agterberg August 21, 2018 Contents 1 Preliminaries 5 1.1 Samples and Population............................. 5
More informationProbability Theory and Statistics. Peter Jochumzen
Probability Theory and Statistics Peter Jochumzen April 18, 2016 Contents 1 Probability Theory And Statistics 3 1.1 Experiment, Outcome and Event................................ 3 1.2 Probability............................................
More informationHomework 10 (due December 2, 2009)
Homework (due December, 9) Problem. Let X and Y be independent binomial random variables with parameters (n, p) and (n, p) respectively. Prove that X + Y is a binomial random variable with parameters (n
More informationExercises and Answers to Chapter 1
Exercises and Answers to Chapter The continuous type of random variable X has the following density function: a x, if < x < a, f (x), otherwise. Answer the following questions. () Find a. () Obtain mean
More informationx k 34 k 34. x 3
A A A A A B A =, C =. f k k x k l f f 3 = k 3 k 3 x k 34 k 34 x 3 k 3 l 3 k 34 l 34. f 4 x 4 K K = [ ] 4 K = K (K) = (K) I m m R I m m = [e, e,, e m ] R = [a, a,, a m ] R = (r ij ) r r r m R = r r m. r
More informationLet X be a continuous random variable, < X < f(x) is the so called probability density function (pdf) if
University of California, Los Angeles Department of Statistics Statistics 1A Instructor: Nicolas Christou Continuous probability distributions Let X be a continuous random variable, < X < f(x) is the so
More informationBrief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More informationTheory of Higher-Order Linear Differential Equations
Chapter 6 Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory A linear differential equation of order n has the form a n (x)y (n) (x) + a n 1 (x)y (n 1) (x) + + a 0 (x)y(x) = b(x), (6.1.1)
More informationMultiple Random Variables
Multiple Random Variables This Version: July 30, 2015 Multiple Random Variables 2 Now we consider models with more than one r.v. These are called multivariate models For instance: height and weight An
More informationAn Introduction to Bessel Functions
An Introduction to R. C. Trinity University Partial Differential Equations March 25, 2014 Bessel s equation Given p 0, the ordinary differential equation x 2 y +xy +(x 2 p 2 )y = 0, x > 0 (1) is known
More informationarxiv: v1 [math.na] 15 Nov 2013
NUMERICAL APPROXIMATIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS arxiv:1311.3935v1 [math.na] 15 Nov 013 Yuri Dimitrov Department of Applied Mathematics and Statistics University of Rousse 8 Studentsa str.
More informationSolution. (i) Find a minimal sufficient statistic for (θ, β) and give your justification. X i=1. By the factorization theorem, ( n
Solution 1. Let (X 1,..., X n ) be a simple random sample from a distribution with probability density function given by f(x;, β) = 1 ( ) 1 β x β, 0 x, > 0, β < 1. β (i) Find a minimal sufficient statistic
More informationECE353: Probability and Random Processes. Lecture 7 -Continuous Random Variable
ECE353: Probability and Random Processes Lecture 7 -Continuous Random Variable Xiao Fu School of Electrical Engineering and Computer Science Oregon State University E-mail: xiao.fu@oregonstate.edu Continuous
More informationMAS223 Statistical Modelling and Inference Examples
Chapter MAS3 Statistical Modelling and Inference Examples Example : Sample spaces and random variables. Let S be the sample space for the experiment of tossing two coins; i.e. Define the random variables
More informationu =0with u(0,x)=f(x), (x) =
PDE LECTURE NOTES, MATH 37A-B 69. Heat Equation The heat equation for a function u : R + R n C is the partial differential equation (.) µ t u =0with u(0,x)=f(x), where f is a given function on R n. By
More informationChp 4. Expectation and Variance
Chp 4. Expectation and Variance 1 Expectation In this chapter, we will introduce two objectives to directly reflect the properties of a random variable or vector, which are the Expectation and Variance.
More informationMA 519 Probability: Review
MA 519 : Review Yingwei Wang Department of Mathematics, Purdue University, West Lafayette, IN, USA Contents 1 How to compute the expectation? 1.1 Tail........................................... 1. Index..........................................
More informationExercises for Chapter 7
Exercises for Chapter 7 Exercise 7.1 The following simple macroeconomic model has four equations relating government policy variables to aggregate income and investment. Aggregate income equals consumption
More informationMeasure-theoretic probability
Measure-theoretic probability Koltay L. VEGTMAM144B November 28, 2012 (VEGTMAM144B) Measure-theoretic probability November 28, 2012 1 / 27 The probability space De nition The (Ω, A, P) measure space is
More informationSecond Order ODEs. Second Order ODEs. In general second order ODEs contain terms involving y, dy But here only consider equations of the form
Second Order ODEs Second Order ODEs In general second order ODEs contain terms involving y, dy But here only consider equations of the form A d2 y dx 2 + B dy dx + Cy = 0 dx, d2 y dx 2 and F(x). where
More informationSummer 2017 MATH Solution to Exercise 5
Summer 07 MATH00 Solution to Exercise 5. Find the partial derivatives of the following functions: (a (xy 5z/( + x, (b x/ x + y, (c arctan y/x, (d log((t + 3 + ts, (e sin(xy z 3, (f x α, x = (x,, x n. (a
More informationRiemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,
Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of
More informationStarting from Heat Equation
Department of Applied Mathematics National Chiao Tung University Hsin-Chu 30010, TAIWAN 20th August 2009 Analytical Theory of Heat The differential equations of the propagation of heat express the most
More informationProbability Density Functions
Probability Density Functions Probability Density Functions Definition Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f (x) such that
More informationINTRODUCTION TO BAYESIAN METHODS II
INTRODUCTION TO BAYESIAN METHODS II Abstract. We will revisit point estimation and hypothesis testing from the Bayesian perspective.. Bayes estimators Let X = (X,..., X n ) be a random sample from the
More informationSOLUTION FOR HOMEWORK 12, STAT 4351
SOLUTION FOR HOMEWORK 2, STAT 435 Welcome to your 2th homework. It looks like this is the last one! As usual, try to find mistakes and get extra points! Now let us look at your problems.. Problem 7.22.
More informationContinuous Distributions
A normal distribution and other density functions involving exponential forms play the most important role in probability and statistics. They are related in a certain way, as summarized in a diagram later
More informationInterval Estimation. Chapter 9
Chapter 9 Interval Estimation 9.1 Introduction Definition 9.1.1 An interval estimate of a real-values parameter θ is any pair of functions, L(x 1,..., x n ) and U(x 1,..., x n ), of a sample that satisfy
More informationHOMEWORK 3 - GEOMETRY OF CURVES AND SURFACES. where ν is the unit normal consistent with the orientation of α (right hand rule).
HOMEWORK 3 - GEOMETRY OF CURVES AND SURFACES ANDRÉ NEVES ) If α : I R 2 is a curve on the plane parametrized by arc length and θ is the angle that α makes with the x-axis, show that α t) = dθ dt ν, where
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More informationProblem Y is an exponential random variable with parameter λ = 0.2. Given the event A = {Y < 2},
ECE32 Spring 25 HW Solutions April 6, 25 Solutions to HW Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where
More informationChapter 3. Second Order Linear PDEs
Chapter 3. Second Order Linear PDEs 3.1 Introduction The general class of second order linear PDEs are of the form: ax, y)u xx + bx, y)u xy + cx, y)u yy + dx, y)u x + ex, y)u y + f x, y)u = gx, y). 3.1)
More informationChapter 2 Continuous Distributions
Chapter Continuous Distributions Continuous random variables For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following
More informationarxiv: v1 [math.co] 6 Jan 2019
BENT FUNCTIONS FROM TRIPLES OF PERMUTATION POLYNOMIALS arxiv:1901.02359v1 [math.co] 6 Jan 2019 DANIELE BARTOLI, MARIA MONTANUCCI, AND GIOVANNI ZINI Abstract. We provide constructions of bent functions
More informationChapter 2 Vector-matrix Differential Equation and Numerical Inversion of Laplace Transform
Chapter 2 Vector-matrix Differential Equation and Numerical Inversion of Laplace Transform 2.1 Vector-matrix Differential Equation A differential equation and a set of differential (simultaneous linear
More informationLecture 5: Moment generating functions
Lecture 5: Moment generating functions Definition 2.3.6. The moment generating function (mgf) of a random variable X is { x e tx f M X (t) = E(e tx X (x) if X has a pmf ) = etx f X (x)dx if X has a pdf
More informationSEVERAL RESULTS OF FRACTIONAL DERIVATIVES IN D (R + ) Chenkuan Li. Author's Copy
RESEARCH PAPER SEVERAL RESULTS OF FRACTIONAL DERIVATIVES IN D (R ) Chenkuan Li Abstract In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on R,
More informationContinuous Random Variables and Continuous Distributions
Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable
More informationFinal Exam # 3. Sta 230: Probability. December 16, 2012
Final Exam # 3 Sta 230: Probability December 16, 2012 This is a closed-book exam so do not refer to your notes, the text, or any other books (please put them on the floor). You may use the extra sheets
More informationa P (A) f k(x) = A k g k " g k (x) = ( 1) k x ą k. $ & g k (x) = x k (0, 1) f k, f, g : [0, 8) Ñ R f k (x) ď g(x) k P N x P [0, 8) g(x)dx g(x)dx ă 8
M M, d A Ď M f k : A Ñ R A a P A f kx = A k xña k P N ta k u 8 k=1 f kx = f k x. xña kñ8 kñ8 xña M, d N, ρ A Ď M f k : A Ñ N tf k u 8 k=1 f : A Ñ N A f A 8ř " x ď k, g k x = 1 k x ą k. & g k x = % g k
More informationXt i Xs i N(0, σ 2 (t s)) and they are independent. This implies that the density function of X t X s is a product of normal density functions:
174 BROWNIAN MOTION 8.4. Brownian motion in R d and the heat equation. The heat equation is a partial differential equation. We are going to convert it into a probabilistic equation by reversing time.
More information1 Random Variable: Topics
Note: Handouts DO NOT replace the book. In most cases, they only provide a guideline on topics and an intuitive feel. 1 Random Variable: Topics Chap 2, 2.1-2.4 and Chap 3, 3.1-3.3 What is a random variable?
More informationIntroduction and preliminaries
Chapter Introduction and preliminaries Partial differential equations What is a partial differential equation? ODEs Ordinary Differential Equations) have one variable x). PDEs Partial Differential Equations)
More informationName: Math Homework Set # 5. March 12, 2010
Name: Math 4567. Homework Set # 5 March 12, 2010 Chapter 3 (page 79, problems 1,2), (page 82, problems 1,2), (page 86, problems 2,3), Chapter 4 (page 93, problems 2,3), (page 98, problems 1,2), (page 102,
More informationA Very Brief Summary of Bayesian Inference, and Examples
A Very Brief Summary of Bayesian Inference, and Examples Trinity Term 009 Prof Gesine Reinert Our starting point are data x = x 1, x,, x n, which we view as realisations of random variables X 1, X,, X
More informationESTIMATION THEORY. Chapter Estimation of Random Variables
Chapter ESTIMATION THEORY. Estimation of Random Variables Suppose X,Y,Y 2,...,Y n are random variables defined on the same probability space (Ω, S,P). We consider Y,...,Y n to be the observed random variables
More informationMath 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3
Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some
More informationBMIR Lecture Series on Probability and Statistics Fall 2015 Discrete RVs
Lecture #7 BMIR Lecture Series on Probability and Statistics Fall 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University 7.1 Function of Single Variable Theorem
More informationExam 1 Review: Questions and Answers. Part I. Finding solutions of a given differential equation.
Exam 1 Review: Questions and Answers Part I. Finding solutions of a given differential equation. 1. Find the real numbers r such that y = e x is a solution of y y 30y = 0. Answer: r = 6, 5 2. Find the
More informationOptimization: Problem Set Solutions
Optimization: Problem Set Solutions Annalisa Molino University of Rome Tor Vergata annalisa.molino@uniroma2.it Fall 20 Compute the maxima minima or saddle points of the following functions a. f(x y) =
More informationMiscellaneous Errors in the Chapter 6 Solutions
Miscellaneous Errors in the Chapter 6 Solutions 3.30(b In this problem, early printings of the second edition use the beta(a, b distribution, but later versions use the Poisson(λ distribution. If your
More informationContinuous random variables
Continuous random variables Continuous r.v. s take an uncountably infinite number of possible values. Examples: Heights of people Weights of apples Diameters of bolts Life lengths of light-bulbs We cannot
More informationOrdinary Differential Equations (ODEs)
Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential
More informationRandom Variables. Cumulative Distribution Function (CDF) Amappingthattransformstheeventstotherealline.
Random Variables Amappingthattransformstheeventstotherealline. Example 1. Toss a fair coin. Define a random variable X where X is 1 if head appears and X is if tail appears. P (X =)=1/2 P (X =1)=1/2 Example
More informationSolution to Assignment 3
The Chinese University of Hong Kong ENGG3D: Probability and Statistics for Engineers 5-6 Term Solution to Assignment 3 Hongyang Li, Francis Due: 3:pm, March Release Date: March 8, 6 Dear students, The
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
More informationTHE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT131 PROBABILITY AND STATISTICS I EXAMPLE CLASS 8 Review Conditional Distributions and Conditional Expectation For any two
More informationHomework 9 (due November 24, 2009)
Homework 9 (due November 4, 9) Problem. The join probability density function of X and Y is given by: ( f(x, y) = c x + xy ) < x
More informationChapter 7. Basic Probability Theory
Chapter 7. Basic Probability Theory I-Liang Chern October 20, 2016 1 / 49 What s kind of matrices satisfying RIP Random matrices with iid Gaussian entries iid Bernoulli entries (+/ 1) iid subgaussian entries
More informationRECHERCHES. sur plusieurs points d analyse relatifs à différens endroits des M. DE LA LAGRANGE
RECHERCHES sur plusieurs points d analyse relatifs à différens endroits des Mémoires précédens. M. DE LA LAGRANGE Mémoires de l Académie Royale des Sciences et Belles-Lettres 179 3 Berlin (1798), pp. 47
More informationS6880 #7. Generate Non-uniform Random Number #1
S6880 #7 Generate Non-uniform Random Number #1 Outline 1 Inversion Method Inversion Method Examples Application to Discrete Distributions Using Inversion Method 2 Composition Method Composition Method
More informationGalois Theory Overview/Example Part 1: Extension Fields. Overview:
Galois Theory Overview/Example Part 1: Extension Fields I ll start by outlining very generally the way Galois theory works. Then, I will work through an example that will illustrate the Fundamental Theorem
More informationHADAMARD OPERATORS ON D (R d )
HADAMARD OPERATORS ON D (R d DIETMAR VOGT BERGISCHE UNIVERSITÄT WUPPERTAL FB MATH.-NAT., GAUSS-STR. 20, D-42119 WUPPERTAL, GERMANY E-MAIL: DVOGT@MATH.UNI-WUPPERTAL.DE Dedicated to the memory of Paweª Doma«ski
More informationVirtuality Distributions and γγ π 0 Transition at Handbag Level
and γγ π Transition at Handbag Level A.V. Radyushkin form hard Physics Department, Old Dominion University & Theory Center, Jefferson Lab May 16, 214, QCD Evolution 214, Santa Fe Transverse Momentum form
More informationSection Let A =. Then A has characteristic equation λ. 2 4λ + 3 = 0 or (λ 3)(λ 1) = 0. Hence the eigenvalues of A are λ 1 = 3 and λ 2 = 1.
Sectio 63 4 3 Let A The A has characteristic equatio λ 2 4λ + 3 or (λ 3)(λ ) Hece the eigevalues of A are λ 3 ad λ 2 λ 3 The correspodig eigevectors satisfy (A λ I 2 )X, or 3 3 or equivaletly x 3y Hece
More informationQualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama
Qualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama Instructions This exam has 7 pages in total, numbered 1 to 7. Make sure your exam has all the pages. This exam will be 2 hours
More informationOn Kummer s distributions of type two and generalized Beta distributions
On Kummer s distributions of type two and generalized Beta distributions Marwa Hamza and Pierre Vallois 2 March 20, 205 Université de Lorraine, Institut de Mathématiques Elie Cartan, INRIA-BIGS, CNRS UMR
More informationStatistics (1): Estimation
Statistics (1): Estimation Marco Banterlé, Christian Robert and Judith Rousseau Practicals 2014-2015 L3, MIDO, Université Paris Dauphine 1 Table des matières 1 Random variables, probability, expectation
More informationApplied Stochastic Models (SS 09)
University of Karlsruhe Institute of Stochastics Prof Dr P R Parthasarathy Dipl-Math D Gentner Applied Stochastic Models (SS 9) Problem Set Problem Let X, X 2 U(θ /2, θ + /2) be independent Show that the
More informationFirst and Second Order ODEs
Civil Engineering 2 Mathematics Autumn 211 M. Ottobre First and Second Order ODEs Warning: all the handouts that I will provide during the course are in no way exhaustive, they are just short recaps. Notation
More informationECE 302 Division 1 MWF 10:30-11:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding.
NAME: ECE 302 Division MWF 0:30-:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding. If you are not in Prof. Pollak s section, you may not take this
More informationIRREDUCIBILITY OF ELLIPTIC CURVES AND INTERSECTION WITH LINES.
IRREDUCIBILITY OF ELLIPTIC CURVES AND INTERSECTION WITH LINES. IAN KIMING 1. Non-singular points and tangents. Suppose that k is a field and that F (x 1,..., x n ) is a homogeneous polynomial in n variables
More information2015 School Year. Graduate School Entrance Examination Problem Booklet. Mathematics
2015 School Year Graduate School Entrance Examination Problem Booklet Mathematics Examination Time: 10:00 to 12:30 Instructions 1. Do not open this problem booklet until the start of the examination is
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationCalculus I (Math 241) (In Progress)
Calculus I (Math 241) (In Progress) The following is a collection of Calculus I (Math 241) problems. Students may expect that their final exam is comprised, more or less, of one problem from each section,
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More information