PhysicsAndMathsTutor.com
|
|
|
- Berenice Cain
- 5 years ago
- Views:
Transcription
1 PhscsAndMathsTutor.com
2 phscsandmathstutor.com June The random varable X has probablt functon k, = 1,, 3, P( X = ) = k ( + 1), = 4, 5, where k s a constant. (a) Fnd the value of k. (b) Fnd the eact value of E(X). (c) Show that, to 3 sgnfcant fgures, Var(X) = (d) Fnd, to 1 decmal place, Var(4 3X). (4) 1 *N0910A010*
3 phscsandmathstutor.com June 005 Queston 5 contnued *N0910A0130* 13 Turn over
4 phscsandmathstutor.com June The random varable X has the dscrete unform dstrbuton P(X = ) = 1, = 1,, 3, 4, 5. 5 (a) Wrte down the value of E(X) and show that Var(X) =. (3) Fnd (b) E(3X ), (c) Var(4 3X). 1 *N337A010*
5 phscsandmathstutor.com Januar The random varable X has probablt functon ( 1) P(X = ) = = 1,, 3, 4, 5, (a) Construct a table gvng the probablt dstrbuton of X. (3) Fnd (b) P( < X 5), (c) the eact value of E(X). (d) Show that Var(X) = 1.97 to 3 sgnfcant fgures. (e) Fnd Var( 3X). (4) 8 *N3957A080*
6 phscsandmathstutor.com Januar 007 Queston 3 contnued Q3 (Total 13 marks) *N3957A090* 9 Turn over
7 phscsandmathstutor.com June The random varable X has probablt dstrbuton P(X = ) 0. p 0. q 0.15 (a) Gven that E(X) = 4.5, wrte down two equatons nvolvng p and q. (3) Fnd (b) the value of p and the value of q, (c) P(4 < X 7). (3) Gven that E(X ) = 7.4, fnd (d) Var(X ), (e) E(19 4X), (1) (f) Var(19 4X). 0 *N6118A004*
8 phscsandmathstutor.com June 007 Queston 7 contnued (Total 13 marks) TOTAL FOR PAPER: 75 MARKS Q7 END *N6118A034* 3
9 phscsandmathstutor.com June The random varable X has probablt dstrbuton gven n the table below P(X = ) p q Gven that E(X) = 0.55, fnd (a) the value of p and the value of q, (b) Var(X), (5) (4) (c) E( X 4). 10 *H358A0108*
10 phscsandmathstutor.com June 008 Queston 3 contnued *H358A0118* 11 Turn over
11 phscsandmathstutor.com Januar When Roht plas a game, the number of ponts he receves s gven b the dscrete random varable X wth the followng probablt dstrbuton P(X = ) (a) Fnd E(X). (b) Fnd F(1.5). (c) Show that Var(X) = 1 (d) Fnd Var(5 3X). (4) Roht can wn a prze f the total number of ponts he has scored after 5 games s at least 10. After 3 games he has a total of 6 ponts. You ma assume that games are ndependent. (e) Fnd the probablt that Roht wns the prze. (6) 10 *N3680A0104*
12 phscsandmathstutor.com Januar 009 Queston 3 contnued *N3680A0114* 11 Turn over
13 phscsandmathstutor.com June The dscrete random varable X has probablt functon P( X = ) = a (3 ) = 0,1, b = 3 (a) Fnd P(X = ) and complete the table below P(X = ) 3a a b (1) Gven that E(X) = 1.6 (b) Fnd the value of a and the value of b. (5) Fnd (c) P(0.5 < X < 3), (d) E(3X ). (e) Show that the Var(X) = 1.64 (3) (f) Calculate Var(3X ). 16 *H3479A0164*
14 phscsandmathstutor.com June 009 Queston 6 contnued *H3479A0174* 17 Turn over
15 phscsandmathstutor.com Januar The probablt functon of a dscrete random varable X s gven b p( ) = k = 1,, 3 where k s a postve constant. (a) Show that 1 k = 14 Fnd (b) P(X ) (c) E(X) (d) Var(1 X) (4) 14 *N35711A0144*
16 phscsandmathstutor.com Januar 010 Queston 5 contnued Q5 (Total 10 marks) *N35711A0154* 15 Turn over
17 phscsandmathstutor.com June The dscrete random varable X has probablt dstrbuton gven b P( X = ) 1 5 a 1 10 a 1 5 where a s a constant. (a) Fnd the value of a. (b) Wrte down E( X ). (c) Fnd Var( X ). (1) (3) The random varable Y = 6 X (d) Fnd Var( Y ). (e) Calculate P( X Y). (3) 8 *H35395A088*
18 phscsandmathstutor.com June 010 Queston 3 contnued *H35395A098* 9 Turn over
19 phscsandmathstutor.com Januar The dscrete random varable X has the probablt dstrbuton P(X = ) k k 3k 4k (a) Show that k = 0.1 (1) Fnd (b) E(X ) (c) E(X ) (d) Var( 5 X ) (3) Two ndependent observatons X 1 and X are made of X. (e) Show that P(X 1 + X = 4) = 0.1 (f) Complete the probablt dstrbuton table for X 1 + X P(X 1 + X = ) (g) Fnd P(1.5 X 1 + X 3.5) 1 *H35410A014*
20 phscsandmathstutor.com Januar 011 Queston 6 contnued *H35410A0134* 13 Turn over
21 phscsandmathstutor.com June 011. The random varable X ~ N ( µ, 5 ) and P( X 3) = (a) Fnd the value of (µ. (4) (b) Wrte down the value of P( µ < X < 3). (1) 4 *P38164A044*
22 phscsandmathstutor.com June The dscrete random varable Y has probablt dstrbuton P(Y = ) a b 0.3 c where a, b and c are constants. The cumulatve dstrbuton functon F() of Y s gven n the followng table F() d 1.0 where d s a constant. (a) Fnd the value of a, the value of b, the value of c and the value of d. (5) (b) Fnd P(3Y + 8). 6 *P38164A064*
23 phscsandmathstutor.com Januar The dscrete random varable X can take onl the values, 3, 4 or 6. For these values the probablt dstrbuton functon s gven b P(X = ) 5 1 where k s a postve nteger. k k 1 (a) Show that k = 3 Fnd (b) F(3) (c) E(X) (d) E ( X ) (1) (e) Var(7 5) (4) 6 *P40699A064*
24 phscsandmathstutor.com Januar 01 Queston 3 contnued *P40699A074* 7 Turn over
25 phscsandmathstutor.com June A dscrete random varable X has the probablt functon P ( X = ) = { k( 1 ) = 1, 0, 1 and 0 otherwse (a) Show that k = 1 6 (b) Fnd E(X) 4 (c) Show that E( X ) = 3 (3) (d) Fnd Var(1 3X) (3) *P40105XA04*
26 phscsandmathstutor.com Januar 013. The dscrete random varable X can take onl the values 1, and 3. For these values the cumulatve dstrbuton functon s defned b 3 + k F( ) = = 13,, 40 (a) Show that k = 13 (b) Fnd the probablt dstrbuton of X. Gven that Var(X) = (4) (c) fnd the eact value of Var(4X 5). 4 *P41805A040*
27 phscsandmathstutor.com Januar A far blue de has faces numbered 1, 1, 3, 3, 5 and 5. The random varable B represents the score when the blue de s rolled. (a) Wrte down the probablt dstrbuton for B. (b) State the name of ths probablt dstrbuton. (c) Wrte down the value of E(B). (1) (1) A second de s red and the random varable R represents the score when the red de s rolled. The probablt dstrbuton of R s r 4 6 P(R = r) (d) Fnd E(R). (e) Fnd Var(R). (3) Tom nvtes Avsha to pla a game wth these dce. Tom spns a far con wth one sde labelled and the other sde labelled 5. When Avsha sees the number showng on the con she then chooses one of the dce and rolls t. If the number showng on the de s greater than the number showng on the con, Avsha wns, otherwse Tom wns. Avsha chooses the de whch gves her the best chance of wnnng each tme Tom spns the con. (f) Fnd the probablt that Avsha wns the game, statng clearl whch de she should use n each case. (4) 14 *P41805A0140*
28 phscsandmathstutor.com Januar 013 Queston 6 contnued *P41805A0150* 15 Turn over
29 phscsandmathstutor.com June 013 (R). The dscrete random varable X takes the values 1, and 3 and has cumulatve dstrbuton functon F() gven b 1 3 F() (a) Fnd the probablt dstrbuton of X. (3) (b) Wrte down the value of F(1.8). (1) Q (Total 4 marks) *P43956A054* 5 Turn over
30 phscsandmathstutor.com June 013 (R) 7. The score S when a spnner s spun has the followng probablt dstrbuton. s P(S = s) (a) Fnd E(S). (b) Show that E(S ) = 10.4 (c) Hence fnd Var(S). (d) Fnd () E(5S 3), () Var(5S 3). (e) Fnd P(5S 3 > S + 3) (4) (3) The spnner s spun twce. The score from the frst spn s S 1 and the score from the second spn s S The random varables S 1 and S are ndependent and the random varable X = S 1 S (f) Show that P({S 1 = 1} X < 5) = 0.16 (g) Fnd P(X < 5). (3) *P43956A04*
31 phscsandmathstutor.com June 013 (R) Queston 7 contnued Q7 (Total 18 marks) END TOTAL FOR PAPER: 75 MARKS 4 *P43956A044*
32 phscsandmathstutor.com June A based de wth s faces s rolled. The dscrete random varable X represents the score on the uppermost face. The probablt dstrbuton of X s shown n the table below P(X = ) a a a b b 0.3 (a) Gven that E(X) = 4. fnd the value of a and the value of b. (b) Show that E(X ) = 0.4 (c) Fnd Var(5 3X) (5) (1) (3) A based de wth fve faces s rolled. The dscrete random varable Y represents the score whch s uppermost. The cumulatve dstrbuton functon of Y s shown n the table below F() k 4k 5k (d) Fnd the value of k. (e) Fnd the probablt dstrbuton of Y. (1) (3) Each de s rolled once. The scores on the two dce are ndependent. (f) Fnd the probablt that the sum of the two scores equals 18 *P4831A0184*
33 phscsandmathstutor.com June 013 Queston 5 contnued *P4831A0194* 19 Turn over
34 Statstcs S1 Probablt P( A B ) = P( A ) + P( B ) P( A B ) P( A B ) = P( A ) P( B A ) P( B A ) P( A ) P( A B ) = P( B A ) P( A ) + P( B A ) P( A ) Dscrete dstrbutons For a dscrete random varable X takng values wth probabltes P(X = ) Epectaton (mean): E(X) = µ = P(X = ) Varance: Var(X) = σ = ( µ ) P(X = ) = P(X = ) µ For a functon g(x ) : E(g(X)) = g( ) P(X = ) Contnuous dstrbutons Standard contnuous dstrbuton: Dstrbuton of X P.D.F. Mean Varance Normal N( µ, σ ) 1 µ 1 σ e σ π µ σ 16 Edecel AS/A level Mathematcs Formulae Lst: Statstcs S1 Issue 1 September 009
35 Edecel AS/A level Mathematcs Formulae Lst: Statstcs S1 Issue 1 September Correlaton and regresson For a set of n pars of values ), ( n S ) ( ) ( = = n S ) ( ) ( = = n S ) )( ( ) )( ( = = The product moment correlaton coeffcent s = = = n n n S S S r ) ( ) ( ) )( ( ) ( ) ( ) )( ( } }{ { The regresson coeffcent of on s ) ( ) )( ( S S b = = Least squares regresson lne of on s b a + = where b a =
36 THE NORMAL DISTRIBUTION FUNCTION The functon tabulated below s Φ(z), defned as Φ(z) = z 1 1 t π e dt. z Φ(z) z Φ(z) z Φ(z) z Φ(z) z Φ(z) Edecel AS/A level Mathematcs Formulae Lst: Statstcs S1 Issue 1 September 009
37 PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION The values z n the table are those whch a random varable Z N(0, 1) eceeds wth probablt p; that s, P(Z > z) = 1 Φ(z) = p. p z p z Edecel AS/A level Mathematcs Formulae Lst: Statstcs S1 Issue 1 September
Expected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
Lecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
Probability and Random Variable Primer
B. Maddah ENMG 622 Smulaton 2/22/ Probablty and Random Varable Prmer Sample space and Events Suppose that an eperment wth an uncertan outcome s performed (e.g., rollng a de). Whle the outcome of the eperment
CS-433: Simulation and Modeling Modeling and Probability Review
CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown
First Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
Edexcel past paper questions
Edexcel past paper questions Statistics 1 Discrete Random Variables Past examination questions Discrete Random variables Page 1 Discrete random variables Discrete Random variables Page 2 Discrete Random
PROBABILITY PRIMER. Exercise Solutions
PROBABILITY PRIMER Exercse Solutons 1 Probablty Prmer, Exercse Solutons, Prncples of Econometrcs, e EXERCISE P.1 (b) X s a random varable because attendance s not known pror to the outdoor concert. Before
Engineering Risk Benefit Analysis
Engneerng Rsk Beneft Analyss.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82, ESD.72, ESD.72 RPRA 2. Elements of Probablty Theory George E. Apostolaks Massachusetts Insttute of Technology Sprng 2007
Correlation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
Q1: Calculate the mean, median, sample variance, and standard deviation of 25, 40, 05, 70, 05, 40, 70.
Q1: Calculate the mean, medan, sample varance, and standard devaton of 5, 40, 05, 70, 05, 40, 70. Q: The frequenc dstrbuton for a data set s gven below. Measurements 0 1 3 4 Frequenc 3 5 8 3 1 a) What
Limited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
Homework 9 for BST 631: Statistical Theory I Problems, 11/02/2006
Due Tme: 5:00PM Thursda, on /09/006 Problem (8 ponts) Book problem 45 Let U = X + and V = X, then the jont pmf of ( UV, ) s θ λ θ e λ e f( u, ) = ( = 0, ; u =, +, )! ( u )! Then f( u, ) u θ λ f ( x x+
Module 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for
Propagation of error for multivariable function
Propagaton o error or multvarable uncton ow consder a multvarable uncton (u, v, w, ). I measurements o u, v, w,. All have uncertant u, v, w,., how wll ths aect the uncertant o the uncton? L tet) o (Equaton
As is less than , there is insufficient evidence to reject H 0 at the 5% level. The data may be modelled by Po(2).
Ch-squared tests 6D 1 a H 0 : The data can be modelled by a Po() dstrbuton. H 1 : The data cannot be modelled by Po() dstrbuton. The observed and expected results are shown n the table. The last two columns
a. (All your answers should be in the letter!
Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal
[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.
![[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.](/thumbs/87/96001374.jpg "[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.")
PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton
Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
A be a probability space. A random vector
Statstcs 1: Probablty Theory II 8 1 JOINT AND MARGINAL DISTRIBUTIONS In Probablty Theory I we formulate the concept of a (real) random varable and descrbe the probablstc behavor of ths random varable by
Introduction to Random Variables
Introducton to Random Varables Defnton of random varable Defnton of random varable Dscrete and contnuous random varable Probablty functon Dstrbuton functon Densty functon Sometmes, t s not enough to descrbe
Hydrological statistics. Hydrological statistics and extremes
5--0 Stochastc Hydrology Hydrologcal statstcs and extremes Marc F.P. Berkens Professor of Hydrology Faculty of Geoscences Hydrologcal statstcs Mostly concernes wth the statstcal analyss of hydrologcal
SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:
SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be
Probability Theory. The nth coefficient of the Taylor series of f(k), expanded around k = 0, gives the nth moment of x as ( ik) n n!
8333: Statstcal Mechancs I Problem Set # 3 Solutons Fall 3 Characterstc Functons: Probablty Theory The characterstc functon s defned by fk ep k = ep kpd The nth coeffcent of the Taylor seres of fk epanded
Statistical analysis using matlab. HY 439 Presented by: George Fortetsanakis
Statstcal analyss usng matlab HY 439 Presented by: George Fortetsanaks Roadmap Probablty dstrbutons Statstcal estmaton Fttng data to probablty dstrbutons Contnuous dstrbutons Contnuous random varable X
A random variable is a function which associates a real number to each element of the sample space
Introducton to Random Varables Defnton of random varable Defnton of of random varable Dscrete and contnuous random varable Probablty blt functon Dstrbuton functon Densty functon Sometmes, t s not enough
xp(x µ) = 0 p(x = 0 µ) + 1 p(x = 1 µ) = µ
CSE 455/555 Sprng 2013 Homework 7: Parametrc Technques Jason J. Corso Computer Scence and Engneerng SUY at Buffalo jcorso@buffalo.edu Solutons by Yngbo Zhou Ths assgnment does not need to be submtted and
Introduction to Regression
Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes
Homework Assignment 3 Due in class, Thursday October 15
Homework Assgnment 3 Due n class, Thursday October 15 SDS 383C Statstcal Modelng I 1 Rdge regresson and Lasso 1. Get the Prostrate cancer data from http://statweb.stanford.edu/~tbs/elemstatlearn/ datasets/prostate.data.
STAT 3340 Assignment 1 solutions. 1. Find the equation of the line which passes through the points (1,1) and (4,5).
(out of 15 ponts) STAT 3340 Assgnment 1 solutons (10) (10) 1. Fnd the equaton of the lne whch passes through the ponts (1,1) and (4,5). β 1 = (5 1)/(4 1) = 4/3 equaton for the lne s y y 0 = β 1 (x x 0
Chapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
Using T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
Linear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
+, where 0 x N - n. k k
CO 745, Mdterm Len Cabrera. A multle choce eam has questons, each of whch has ossble answers. A student nows the correct answer to n of these questons. For the remanng - n questons, he checs the answers
THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for
Statistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
Applied Stochastic Processes
STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of
Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
Rockefeller College University at Albany
Rockefeller College Unverst at Alban PAD 705 Handout: Maxmum Lkelhood Estmaton Orgnal b Davd A. Wse John F. Kenned School of Government, Harvard Unverst Modfcatons b R. Karl Rethemeer Up to ths pont n
Chapter 1. Probability
Chapter. Probablty Mcroscopc propertes of matter: quantum mechancs, atomc and molecular propertes Macroscopc propertes of matter: thermodynamcs, E, H, C V, C p, S, A, G How do we relate these two propertes?
Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
Strong Markov property: Same assertion holds for stopping times τ.
Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up
MOUNT SAINT JOSEPH MILPERRA
Student Name : MOUNT SAINT JOSEPH MILPERRA MOUNT SAINT JOSEPH I YEAR 8 MATHEMATICS 17th November 2006 70 MINUTES (plus 5 mnutes readng tme)... :... INSTRUCTIONS: > WRITE IN BLUE OR BLACK PEN ONLY > ALL
Maximum Likelihood Estimation
Maxmum Lkelhood Estmaton INFO-2301: Quanttatve Reasonng 2 Mchael Paul and Jordan Boyd-Graber MARCH 7, 2017 INFO-2301: Quanttatve Reasonng 2 Paul and Boyd-Graber Maxmum Lkelhood Estmaton 1 of 9 Why MLE?
ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
Statistics and Probability Theory in Civil, Surveying and Environmental Engineering
Statstcs and Probablty Theory n Cvl, Surveyng and Envronmental Engneerng Pro. Dr. Mchael Havbro Faber ETH Zurch, Swtzerland Contents o Todays Lecture Overvew o Uncertanty Modelng Random Varables - propertes
LECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
MIMA Group. Chapter 2 Bayesian Decision Theory. School of Computer Science and Technology, Shandong University. Xin-Shun SDU
Group M D L M Chapter Bayesan Decson heory Xn-Shun Xu @ SDU School of Computer Scence and echnology, Shandong Unversty Bayesan Decson heory Bayesan decson theory s a statstcal approach to data mnng/pattern
January Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
Lecture 17 : Stochastic Processes II
: Stochastc Processes II 1 Contnuous-tme stochastc process So far we have studed dscrete-tme stochastc processes. We studed the concept of Makov chans and martngales, tme seres analyss, and regresson analyss
Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
Math1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
Linear regression. Regression Models. Chapter 11 Student Lecture Notes Regression Analysis is the
Chapter 11 Student Lecture Notes 11-1 Lnear regresson Wenl lu Dept. Health statstcs School of publc health Tanjn medcal unversty 1 Regresson Models 1. Answer What Is the Relatonshp Between the Varables?.
Modeling and Simulation NETW 707
Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C3.220 1 Propertes of Random Numbers Random Number Generators (RNGs) must
Lecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
Stat 543 Exam 2 Spring 2016
Stat 543 Exam 2 Sprng 2016 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of 11 questons. Do at least 10 of the 11 parts of the man exam.
Statistics MINITAB - Lab 2
Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that
Complex Numbers Alpha, Round 1 Test #123
Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test
CALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules
Stat 543 Exam 2 Spring 2016
Stat 543 Exam 2 Sprng 206 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of questons. Do at least 0 of the parts of the man exam. I wll score
Professor Chris Murray. Midterm Exam
Econ 7 Econometrcs Sprng 4 Professor Chrs Murray McElhnney D cjmurray@uh.edu Mdterm Exam Wrte your answers on one sde of the blank whte paper that I have gven you.. Do not wrte your answers on ths exam.
The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
Differentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
Scatter Plot x
Construct a scatter plot usng excel for the gven data. Determne whether there s a postve lnear correlaton, negatve lnear correlaton, or no lnear correlaton. Complete the table and fnd the correlaton coeffcent
Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of
Chapter 7 Generalzed and Weghted Least Squares Estmaton The usual lnear regresson model assumes that all the random error components are dentcally and ndependently dstrbuted wth constant varance. When
Multiple Choice. Choose the one that best completes the statement or answers the question.
ECON 56 Homework Multple Choce Choose the one that best completes the statement or answers the queston ) The probablty of an event A or B (Pr(A or B)) to occur equals a Pr(A) Pr(B) b Pr(A) + Pr(B) f A
COMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE School of Computer and Communcaton Scences Handout 0 Prncples of Dgtal Communcatons Solutons to Problem Set 4 Mar. 6, 08 Soluton. If H = 0, we have Y = Z Z = Y
MATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
The written Master s Examination
he wrtten Master s Eamnaton Opton Statstcs and Probablty SPRING 9 Full ponts may be obtaned for correct answers to 8 questons. Each numbered queston (whch may have several parts) s worth the same number
Limited Dependent Variables and Panel Data. Tibor Hanappi
Lmted Dependent Varables and Panel Data Tbor Hanapp 30.06.2010 Lmted Dependent Varables Dscrete: Varables that can take onl a countable number of values Censored/Truncated: Data ponts n some specfc range
Chapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
AS-Level Maths: Statistics 1 for Edexcel
1 of 6 AS-Level Maths: Statstcs 1 for Edecel S1. Calculatng means and standard devatons Ths con ndcates the slde contans actvtes created n Flash. These actvtes are not edtable. For more detaled nstructons,
Suites of Tests. DIEHARD TESTS (Marsaglia, 1985) See
Sutes of Tests DIEHARD TESTS (Marsagla, 985 See http://stat.fsu.edu/~geo/dehard.html NIST Test sute- 6 tests on the sequences of bts http://csrc.nst.gov/rng/ Test U0 Includes the above tests. http://www.ro.umontreal.ca/~lecuyer/
Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
b ), which stands for uniform distribution on the interval a x< b. = 0 elsewhere
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Some mportant probablty dstrbutons: Unform Bnomal Posson Gaussan/ormal The Unform dstrbuton s often called U( a, b ), hch stands for unform
e i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)
ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston
Definition. Measures of Dispersion. Measures of Dispersion. Definition. The Range. Measures of Dispersion 3/24/2014
Measures of Dsperson Defenton Range Interquartle Range Varance and Standard Devaton Defnton Measures of dsperson are descrptve statstcs that descrbe how smlar a set of scores are to each other The more
Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
A REVIEW OF ERROR ANALYSIS
A REVIEW OF ERROR AALYI EEP Laborator EVE-4860 / MAE-4370 Updated 006 Error Analss In the laborator we measure phscal uanttes. All measurements are subject to some uncertantes. Error analss s the stud
Negative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
Marginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients
ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It
CSCE 790S Background Results
CSCE 790S Background Results Stephen A. Fenner September 8, 011 Abstract These results are background to the course CSCE 790S/CSCE 790B, Quantum Computaton and Informaton (Sprng 007 and Fall 011). Each
Measuring the Strength of Association
Stat 3000 Statstcs for Scentsts and Engneers Measurng the Strength of Assocaton Note that the slope s one measure of the lnear assocaton between two contnuous varables t tells ou how much the average of
Biostatistics. Chapter 11 Simple Linear Correlation and Regression. Jing Li
Bostatstcs Chapter 11 Smple Lnear Correlaton and Regresson Jng L jng.l@sjtu.edu.cn http://cbb.sjtu.edu.cn/~jngl/courses/2018fall/b372/ Dept of Bonformatcs & Bostatstcs, SJTU Recall eat chocolate Cell 175,
Interval Regression with Sample Selection
Interval Regresson wth Sample Selecton Géraldne Hennngsen, Arne Hennngsen, Sebastan Petersen May 3, 07 Ths vgnette s largely based on Petersen et al. 07. Model Specfcaton The general specfcaton of an nterval
Solution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
Credit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
6.4. RANDOM VARIABLES 233
6.4. RANDOM VARIABLES 2 6.4 Random Varables What are Random Varables? A random varable for an experment wth a sample space S s a functon that assgns a number to each element of S. Typcally nstead of usng
The Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
Beyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
Answers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%