Formulas in Project Risk
|
|
- Ὕδρα Αναγνωστάκης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Formulas in Project Risk Jørn Vatn Rev2 Some important formulas from the course compendium Project Risk Analysis are listed in this memo. For assumptions and limitations, please consult the compendium. Denne formelsamlingen er tillatt hjelpemiddel under eksamen i TPK 5115 Risikostyring i prosjekter, 6. desember Studentene kan skrive notater i formelsamlingen og på omslaget. Formelsamlingen inneholder 16 sider og skal skrives ut med skrift på begge sider. The table of formulas can be used during the exam in TPK 5115 Risk Management in Projects, December 6th, The students are allowed to make notes in the memo and on its cover. The table of formula contains 16 pages and shall be printed with double sided print. 1
2 Chapter 3 Basic Probability Rules Pr(A B) = Pr(A) + Pr(B) Pr(A B) Pr(A B) = Pr(A) Pr(B) if A and B are independent Pr(A C ) = Pr(A does not occur) = 1 Pr(A) Pr(A B) Pr(A B) = Pr(B) The Law of Total Probability Bayes Theorem r Pr(B) = Pr(A i ) Pr(B A i ) i=1 Pr(A j B) = Pr(B A j) Pr(A j ) r Pr(A i ) Pr(B A i ) i=1 Cumulative Distribution Function - CDF F X (x) = Pr(X x) Pr(a < X b) = F X (b) F X (a) Probability Density Function - PDF f X (x) = d dx F X (x) F X (x) = x f X (u)du b Pr(a < X b) = f X (x)dx a 2
3 Point Probability p(x j ) = Pr(X = x j ) Expectation Variance x f X (x) dx if X is continuous E(X) = x j p(x j ) if X is discrete j [x E(X)] 2 f X (x) dx if X is continuous Var(X) = [ (x j E(X) ] 2 p(x j ) if X is discrete Standard Deviation j SD(X) = + Var(X) Double Expectation Rules E(X) = E(E(X Y )) Var(X) = E(Var(X Y )) + Var(E(X Y )) E(X) = E(X B)Pr(B) + E(X B C )Pr(B C ) Var(X) = Var(X B)Pr(B) + Var(X B C )Pr(B C ) [ 2 +[E(X B) E(X)] 2 Pr(B) + E(X B C ) E(X)] Pr(B C ) Normal Distribtuion f X (x) = 1 1 (x µ) 2 2π σ e 2σ 2 E(X) = µ Var(X) = σ 2 3
4 Standard Normal Distribtuion f U (u) = φ(u) = 1 e u2 2 2π F U (u) = Φ(u) = u φ(t)dt = u 1 2π e t2 2 dt Transforming to the Standard Normal Distribution Exponential Distribtuion U = X µ σ f X (x) = λe λx F X (x) = 1 e λx Weibull Distribution E(X) = 1/λ Var(X) = 1/λ 2 f X (x) = αλ(λx) α 1 e (λx)α Gamma Distribution F X (x) = 1 e (λx)α E(X) = 1 [ ) 1 λ Γ α + 1 Var(X) = 1 ( ) ( )] 2 1α (Γ λ 2 α + 1 Γ f X (x) = λα Γ(α) (x)α 1 e λx 4
5 Erlang Distribution: α = integer F X (x) = 1 α 1 n=0 (λx) α Gamma- and Erlang Distribution - Moments: n! e (λx) E(X) = α λ Var(X) = α λ 2 Inverted Gamma Distribution Lognormal Distribution f X (x) = λα Γ(α) ( ) 1 α+1 e λ/x x E(X) = λ/(α 1) Var(X) = λ 2 (α 1) 2 (α 2) 1 f X (x) = π τ x e 1 2τ2 (log x ν)2 Binomial Distribution ( n Pr(X = x) = x E(X) = e ν+ 1 2 τ2 Var(X) = e 2ν (e 2τ2 e τ2 ) ) p x (1 p ) n x for x = 1,2,.., n Poisson Distribution E(X) = np Var(X) = np(1 p) p(x) = Pr(X = x) = λx x! e λ E(X) = λ Var(X) = λ 5
6 Poisson Process: Number of Events in an Interval p(x) = Pr(X = x) = Inverse-Gauss Distribution ( λ F T (t) = Φ µ Triangular Distribution f X (x) = F X (x) = PERT Distribution t λ 1 t ) + Φ { { [λ(b a)]x e λ(b a) x! E(T) = µ Var(T) = µ 3 /λ 2(x L) (M L)(H L) 2(H x) (H M)(H L) (x L) 2 (M L)(H L) 1 (H x)2 (H M)(H L) ( λ ) 1 t λ t e 2λ/µ µ if L x M if M x H if L x M if M x H E(X) = L + M + H 3 Var(X) = L2 + M 2 + H 2 LM LH MH 18 Introduce: Then 4M + H 5L α 1 = H L 5H 4M L α 2 = H L z = x L H L f X (x) = (x L)α 1 1 (H x) α 2 1 B(α 1,α 2 )(H L) α 1+α 2 1 F X (x) = B z(α 1,α 2 ) B(α 1,α 2 ) E(X) = L + 4M + H 6 (E(X) L)(H E(X)) Var(X) = 7 6
7 Distribution of a Sum E(X 1 + X X n ) = E ( n i=1 X i) = n i=1 E(X i) Variables are independent: Var(X 1 + X X n ) = Var ( n i=1 X i) = n i=1 Var(X i) Variables are dependent (n = 2): SD ( n i=1 X ) n i = i=1 [SD(X i)] 2 Var(X 1 + X 2 ) = Var(X 1 ) + Var(X 2 ) + 2Cov(X 1, X 2 ) Distribution of a Product Variables are independent: ( ) n n E(X 1 X 2... X n ) = E X i = E(X i ) i=1 i=1 Var(X 1 X 2 ) = Var(X 1 )Var(X 2 ) + Var(X 1 )[E(X 2 )] 2 + Var(X 2 )[E(X 1 )] 2 SD(X 1 X 2 ) = Var(X 1 )Var(X 2 ) + Var(X 1 )[E(X 2 )] 2 + Var(X 2 )[E(X 1 )] 2 Distribution of maximum values Let Y = max(x 1, X 2 ): F Y (x) = Pr(Y x) = Pr(X 1 x X 2 x) = Pr(X 1 x)pr(x 2 x) = F X1 (x)f X2 (x) E(Y ) = x f Y (x) dx = x [f X1 (x)f X2 (x) + f X2 (x)f X1 (x) ] dx Var(Y ) = [x E(Y )] 2 [f X1 (x)f X2 (x) + f X2 (x)f X1 (x) ] dx 7
8 Chapter 4 Total Expected Penalty for Default Chapter 5 PDTot = D (t D)PDf T (t) dt Fundamental Utility Function Requirements y 1 y 0 y 2, and Y = y 0 is the certain outcome: u(y 0 ) = αu(y 1 ) + (1 α)u(y 2 ) Typical utility function dealing with safety attributes: u(y 1, y 2, y 3, y 4 ) = 0.03y 1 0.5y 2 2.5y 3 7y 4 Typical utility function dealing with safety attributes and profit: u(y 1, y 2, y 3, y 4 ) = 0.03y 1 0.5y 2 2.5y 3 7y 4 + y 7 ae by 7 Chapter 6 NPV Formulas One amount: NPV = X t (1 + r) t Cash Flow: T NPV = X t (1 + r) t t=0 Fixed Amount: Increasing Amount: [ 1 (1 + r) T NPV = r NPV = 1 ( 1+v 1+r r v ) T ] X A X A,v 8
9 Periodic Amount NPV = X A (1 + r) ki X A = 1 (1 + r) k i=0 Periodic amount when the first amount occurs at the end of year l: Degradation Cost in year t in case of degradation NPV = X A(1 + r) l 1 (1 + r) k c t = c 0 (1 + d) t Degradation rate found from growth factor: d = e ln(gf)/t 1 Chapter 7 Maximum Likelihood Principle Simultaneous probability density n f (x 1 ;θ)f (x 2 ;θ)... f (x n ;θ) = f (x i ;θ) i=1 Likelihood function n L(θ; x 1, x 1... x n ) = f (x i ;θ) i=1 MLE ˆθ = ˆθ(X1, X 2,... X n ) Methods of Moments - PERT Distribution ˆM = ¼(6 x ˆL Ĥ) Ĥ = x + S 7 x Max x x x Min ˆL = x (Ĥ x)( x x Min ) x Max x 9
10 LS Principle Simple linear regression model: n Q(θ) = [y i φ i (θ] 2 i=1 Multiple linear model: E(Y i ) = β 0 + β 1 x i E(Y i ) = β 0 + β 1 x i,1 + β 2 x i,2 + + β r x i,r X = 1 x x 1r 1 x 21 x 1r : x i j 1 x i1... x nr X T y = X T Xβ With error terms: ˆβ = (X T X) 1 X T y Predictions: Residuals: Y i = β 0 + β 1 x i,1 + β 2 x i,2 + + β r x i,r + ε i ŷ i = ˆ β 0 + ˆ β 1 x i,1 + ˆ β 2 x i,2 + + ˆ β r x i,r εˆ i = y i ŷ i Chapter 8 Calibration Let Z be the number of Y i x i that are >0 For n 5, a calibration is done when Z <n/2 - n or Z >n/2 + n For 2 n 4 calibrate if Z = 0 or Z = n, and: (i) Z = 0 and 1/n i(y i /x i ) <1/(6-n), or (ii) Z = n and 1/n i(y i /x i ) >(6-n) 10
11 Calibration Formulas Relation between the true values (x i s) and the estimates (Y i s): x i = β 0 + β 1 Y i + error term LS formulas: i (Y i Ȳ )x i ˆβ 1 = ( Yi Ȳ ) 2 i ˆβ 0 = x ˆβ1 Ȳ New value: Regression line through the origin: Weighting of Experts Error terms from control questions: ˆx = ˆβ0 + ˆβ1 y ( ) ˆβ 0 ˆx = + ˆβ1 x i,min Y i = α 0 + α 1 x i + error term Square sum of the residuals for expert k: Estimate for the variance: Standardised weight of expert k: SS k = i (y i αˆ 0 ˆα 1 x i ) 2 S 2 k = SS k/(n 2) w k = S 2 k j S 2 j Weight for expert k based on mutual evaluation: w k = j j p j,k y i j p j,i Standard weighting model - Experts only: ˆx = j=1:m w j ˆx j 11
12 Experts and data Sample variance from expert statements: S 2 V E = 1 1 m j=1 w2 j Self evaluated standard deviation : Variance of the weighted estimate: m j=1 Ŝ k = 0.37( ˆx k,h ˆx k,l ) S 2 SE = 1 j=1:m Ŝ 2 j Combined estimate - Experts and Data: Chapter 9 ˆx = S 2 E ˆx E + S 2 D ˆx D S 2 E + S 2 D ( w j xˆ j ˆx ) 2 Single Component Maintenance Models C(τ) = C PM /τ + λ E (τ)[c CM + C EP + C ES ] Effective failure rate approximation: ( ) Γ(1 + 1/α) α λ E (τ) = τ α 1 MTTF Improved approximation: ( ) Γ(1 + 1/α) α λ E (τ) = τ α 1 [ 1 0.1α(τ/MTTF) 2 + (0.09α 0.2)τ/MTTF ] MTTF Optimal interval in the simple model: τ = MTTF ( C PM Γ(1 + 1/α) C U (α 1) ) 1/α Single activity - Dynamic considerations C First TA = C TA + λ E (τ TA ) C U τ TA C Second TA = λ E (τ TA + x) C U (τ TA + x) λ E (x) C U x 12
13 Single activity - Artic maintenance C First TA = C TA + [1 R(τ TA )]C U + τta t=0 f T (t)(τ TA t)c W dt τta C Second TA = [1 R(τ TA + x)/r(x)]c U + f T (t + x)(τ TA t)c W dt/r(x) t=0 Random cost due to delaying the turnaround: E(C T A,R ) = C PL t=d TA [f TI (t) f TO (t)](t D TA ) dt Total cost of PM when included in the turnaround if turnaround may be dealyed: C TA = C TA,F + E(C TA,R ) Possibilities to cancel PM from turnaround: PM outside TA C 1 = C PM + C U [ λe (τ )τ λ E (τ TA )τ TA ] + C (2τ TA τ ) PM planned in TA, but cancelled C 2 = C PM + C TA,P + C U [ λe (τ )τ λ E (τ TA )τ TA ] + C (2τ TA τ ) +C PL t=d TA f TO (t)(t D TA )dt PM in TA & problems C 3 = C TA,E + C TA,P + C U λ E (τ TA )τ TA + C PL PM in TA without any problems C 4 = C TA,E + C TA,P + C U λ E (τ TA )τ TA + C PL Changing the frequency of the turnaround Assuming uppgrade of safety critical components: C(τ TA ) = C τta,b /τ TA + n i C TA,i /τ TA + i {SC} t=d TA f TI (t)(t D TA )dt t=d TA f TN (t)(t D TA )dt i {NC} n i C U,i λ E,i (τ TA ) Yearly upgrading cost: C UG = n i C UG,i /T i {SC} 13
14 Discounted number of failures τ TA ( Λ E,i (τ TA, r) = jλe,i ( j) ( j 1)λ E,i ( j 1) ) (1 + r) j j=1 The turnaround related cost up to time of disposal: C A (τ TA = ( n(t,τta ) j=1 (1 + r) ( j 1)τ TA C TA,B + i {SC} n ic TA,i + ) i {NC} n ic U,i Λ E,i (τ TA, r) + C FLP + i {SC} n ic UG,i Cost related to exclude production related activities from the turnaround: n(t,τ i ) (1 + r) ( j 1)τ i j=1 The Gamma Function ( ni C PM,i + n i C U,i Λ E,i ( τ i, r)) The gamma function Γ(α) is defined for all real α > 0 by the integral Γ(α) = 0 t α 1 e t dt By partial integration it is easy to show that Γ(α + 1) = αγ(α) for all α > 0 (1) In Table 2 the Gamma function Γ(α) is given for values of α between 1.00 and Γ(α) for other positive values of α may be calculated from formula (1). 14
15 Table 1: The Cumulative Standard Normal Distribution Φ(z) = Pr(Z z) = z 1 2π e u2 2 du z Φ(-z) = 1 - Φ(z) 15
16 Table 2: Gamma Function Γ(α) for α between 1.00 and α Γ(α) α Γ(α) α Γ(α) α Γ(α)
FORMULAS FOR STATISTICS 1
FORMULAS FOR STATISTICS 1 X = 1 n Sample statistics X i or x = 1 n x i (sample mean) S 2 = 1 n 1 s 2 = 1 n 1 (X i X) 2 = 1 n 1 (x i x) 2 = 1 n 1 Xi 2 n n 1 X 2 x 2 i n n 1 x 2 or (sample variance) E(X)
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραStatistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραAn Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Διαβάστε περισσότεραAPPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραBiostatistics for Health Sciences Review Sheet
Biostatistics for Health Sciences Review Sheet http://mathvault.ca June 1, 2017 Contents 1 Descriptive Statistics 2 1.1 Variables.............................................. 2 1.1.1 Qualitative........................................
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραProbability and Random Processes (Part II)
Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραBayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
Διαβάστε περισσότεραHOMEWORK#1. t E(x) = 1 λ = (b) Find the median lifetime of a randomly selected light bulb. Answer:
HOMEWORK# 52258 李亞晟 Eercise 2. The lifetime of light bulbs follows an eponential distribution with a hazard rate of. failures per hour of use (a) Find the mean lifetime of a randomly selected light bulb.
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραAquinas College. Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET
Aquinas College Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical
Διαβάστε περισσότερα255 (log-normal distribution) 83, 106, 239 (malus) 26 - (Belgian BMS, Markovian presentation) 32 (median premium calculation principle) 186 À / Á (goo
(absolute loss function)186 - (posterior structure function)163 - (a priori rating variables)25 (Bayes scale) 178 (bancassurance)233 - (beta distribution)203, 204 (high deductible)218 (bonus)26 ( ) (total
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραHomework for 1/27 Due 2/5
Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where
Διαβάστε περισσότεραWritten Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Διαβάστε περισσότερα1. A fully continuous 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1
Chapter 7: Exercises 1. A fully continuous 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1 35+n:30 n a 35+n:20 n 0 0.068727 11.395336 10 0.097101 7.351745 25
Διαβάστε περισσότεραMain source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
Διαβάστε περισσότεραEstimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University
Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότερα1 1 1 2 1 2 2 1 43 123 5 122 3 1 312 1 1 122 1 1 1 1 6 1 7 1 6 1 7 1 3 4 2 312 43 4 3 3 1 1 4 1 1 52 122 54 124 8 1 3 1 1 1 1 1 152 1 1 1 1 1 1 152 1 5 1 152 152 1 1 3 9 1 159 9 13 4 5 1 122 1 4 122 5
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραAssalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραSupplementary Appendix
Supplementary Appendix Measuring crisis risk using conditional copulas: An empirical analysis of the 2008 shipping crisis Sebastian Opitz, Henry Seidel and Alexander Szimayer Model specification Table
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραFundamentals of Probability: A First Course. Anirban DasGupta
Fundamentals of Probability: A First Course Anirban DasGupta Contents 1 Introducing Probability 5 1.1 ExperimentsandSampleSpaces... 6 1.2 Set Theory Notation and Axioms of Probability........... 7 1.3
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραExercises to Statistics of Material Fatigue No. 5
Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραLecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
Διαβάστε περισσότεραIntroduction to the ML Estimation of ARMA processes
Introduction to the ML Estimation of ARMA processes Eduardo Rossi University of Pavia October 2013 Rossi ARMA Estimation Financial Econometrics - 2013 1 / 1 We consider the AR(p) model: Y t = c + φ 1 Y
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραMatrices and vectors. Matrix and vector. a 11 a 12 a 1n a 21 a 22 a 2n A = b 1 b 2. b m. R m n, b = = ( a ij. a m1 a m2 a mn. def
Matrices and vectors Matrix and vector a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn def = ( a ij ) R m n, b = b 1 b 2 b m Rm Matrix and vectors in linear equations: example E 1 : x 1 + x 2 + 3x 4 =
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότερα6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραAdditional Results for the Pareto/NBD Model
Additional Results for the Pareto/NBD Model Peter S. Fader www.petefader.com Bruce G. S. Hardie www.brucehardie.com January 24 Abstract This note derives expressions for i) the raw moments of the posterior
Διαβάστε περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραHW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)
HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραP AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Διαβάστε περισσότεραQueensland University of Technology Transport Data Analysis and Modeling Methodologies
Queensland University of Technology Transport Data Analysis and Modeling Methodologies Lab Session #7 Example 5.2 (with 3SLS Extensions) Seemingly Unrelated Regression Estimation and 3SLS A survey of 206
Διαβάστε περισσότεραA Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραThe ε-pseudospectrum of a Matrix
The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems
Διαβάστε περισσότεραΥπολογιστική Φυσική Στοιχειωδών Σωματιδίων
Υπολογιστική Φυσική Στοιχειωδών Σωματιδίων Όρια Πιστότητας (Confidence Limits) 2/4/2014 Υπολογ.Φυσική ΣΣ 1 Τα όρια πιστότητας -Confidence Limits (CL) Tα όρια πιστότητας μιας μέτρησης Μπορεί να αναφέρονται
Διαβάστε περισσότεραDurbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
Διαβάστε περισσότεραLecture 7: Overdispersion in Poisson regression
Lecture 7: Overdispersion in Poisson regression Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen 2004 0 Overview Introduction Modeling overdispersion through mixing Score test for
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραModule 5. February 14, h 0min
Module 5 Stationary Time Series Models Part 2 AR and ARMA Models and Their Properties Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W. Q. Meeker. February 14,
Διαβάστε περισσότεραΑλγόριθμοι και πολυπλοκότητα NP-Completeness (2)
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2) Ιωάννης Τόλλης Τμήμα Επιστήμης Υπολογιστών NP-Completeness (2) x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 11 13 21
Διαβάστε περισσότεραg-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις
Διαβάστε περισσότεραdepartment listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι
She selects the option. Jenny starts with the al listing. This has employees listed within She drills down through the employee. The inferred ER sttricture relates this to the redcords in the databasee
Διαβάστε περισσότεραAnti-Final CS/SE 3341 SOLUTIONS
CS/SE 3341 SOLUTIONS Anti-Final 1. Users call help desk every 15 minutes, on the average. There is one help desk specialist on duty, and her average service time is 9 minutes. Modeling the help desk as
Διαβάστε περισσότεραDynamic types, Lambda calculus machines Section and Practice Problems Apr 21 22, 2016
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Dynamic types, Lambda calculus machines Apr 21 22, 2016 1 Dynamic types and contracts (a) To make sure you understand the
Διαβάστε περισσότεραSECTION II: PROBABILITY MODELS
SECTION II: PROBABILITY MODELS 1 SECTION II: Aggregate Data. Fraction of births with low birth weight per province. Model A: OLS, using observations 1 260 Heteroskedasticity-robust standard errors, variant
Διαβάστε περισσότεραMean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραη π 2 /3 χ 2 χ 2 t k Y 0/0, 0/1,..., 3/3 π 1, π 2,..., π k k k 1 β ij Y I i = 1,..., I p (X i = x i1,..., x ip ) Y i J (j = 1,..., J) x i Y i = j π j (x i ) x i π j (x i ) x (n 1 (x),..., n J (x))
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ
ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότερα5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότερα22 .5 Real consumption.5 Real residential investment.5.5.5 965 975 985 995 25.5 965 975 985 995 25.5 Real house prices.5 Real fixed investment.5.5.5 965 975 985 995 25.5 965 975 985 995 25.3 Inflation
Διαβάστε περισσότεραΤΟ ΜΟΝΤΕΛΟ Οι Υποθέσεις Η Απλή Περίπτωση για λi = μi 25 = Η Γενική Περίπτωση για λi μi..35
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΣΤΑΤΙΣΤΙΚΗΣ ΚΑΙ ΕΠΙΧΕΙΡΗΣΙΑΚΗΣ ΕΡΕΥΝΑΣ ΑΝΑΛΥΣΗ ΤΩΝ ΣΥΣΧΕΤΙΣΕΩΝ ΧΡΕΟΚΟΠΙΑΣ ΚΑΙ ΤΩΝ
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότερα