GEOMETRIC discrete A discrete random variable R counts number of times needed before an event occurs


 Hilary McDowell
 1 years ago
 Views:
Transcription
1 STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,... where 0 < p < GIVEN IN FORMULA BOOK Conditions/Assumptions : there is a sequence of independent trials only outcomes, success and failure, constant probability, p of success at each trial a For P(X a) or P(X a) use the sum of a GP to evaluate S =  r X~Geo(0.5) Find P(X>5) P(X>5) = P(X=6)+P(X=7)+P(X=8).. = GP. a = 0.5 x r= 0.75 P(X>5) = = Mean of Geo(p) = p Variance of Geo(p) = mean x (mean ) MAKE SURE YOU CAN WRITE OUT FULLY THE PROOF FOR THESE!!! EXPONENTIAL continuous Intervals of time between events occurring related to the Poisson if the number of events occurring in a given period of time is Poisson then the time between successive events is exponential constant probability of an even occurring per unit of time. f(x) = le lx x ³ 0 0 x < 0 l = average time between events GIVEN IN FORMULA BOOK MEAN = E(X) = VARIANCE = l l For P(X b) or P(X b) use the distribution function P(X b) = b ò 0 le lx dx = [ e lx ] b 0 = e lb Can be shown using integration by parts
2 Important Feature exponential is a memoryless distribution important for conditional probability. The probability that we need to wait more tha0 more seconds for the first event to occur given that it has not happened after waiting 30 seconds, is the same as the probability that we need to wait more tha0 seconds.. Estimation MAKE SURE YOU LEARN THE PROOF for E(S ) = s (Page 0) A statistic used to estimate the value of a parameter of a population is called an estimator The Most Efficient estimator is the one which o is unbiased it s expected value = the parameter it is estimating o has the smallest variance. Consistent Estimator : If U is an unbiased estimator for an unknown parameter θ, then U is a consistent estimator for θ if Var (U) 0 as n, where n is the size of the sample  you may need to use Σr, Σr, Σr 3  all given in formula booklet Relative Efficiency of Estimator A to Estimator B = = / Var(Estimator A) / Var(Estimator B) A random variable X has mean µ and variance 0 A random variable Y has mean µ and variance 5 a) Given that ax + by is and unbiased estimator of µ, show that a + b=. E(aX + by) = m ae(x) + be(y) = m ma + mb = m a + b = b) The variance of ax + by is denoted by V. Express V in the form pa + qa + r Var(aX + by) = a Var(X) + b Var(Y) = 0a + 5b =0a + 5(a ) =30a 0a + 5 c) Find the values of a and b such that V takes its minimum value. Method Differentiation Method Completing the square dv = 60a 0 30 æ da è ç a ö 3a + 5 ø 60a 0 = 0 a = 3 so b = 3 30 æ è ç a ö 3 ø 30 æ ç ö è 3 ø + 5 a = 3 so b = 3 d) A single observation is taken on each of X and Y. The values observed are 0 and 6 respectively. Use results from c) to estimate µ. m = E(X) + E(Y) = = 5 3
3 Estimator of a Population Proportion (Binomial) From a binomial population which p, is the proportion of successes (unknown), a random sample of size n is taken. X is the number of successes P s is the proportion of successes in the sample P s = X n P s is an unbiased estimator for p as æ E X ö E(P s ) = ç = E(X) = è n ø n n (np) = p Mean of a binomial E(X) = np Var(P s ) = Var æ è ç X ö n ø = n Var(X) = p( p) (np( p)) = n n Pooled estimators of Population Proportions Size Unbiased estimator Of popn proportion Variance of a binomial Var(X) = np(  p) Proportion Sample I P s Sample II n P s p = P s + n P s + n E æ n P è ç + n P ö s s + n = [E( P s ) + E(n P s )] ø + n = = = p + n [ E(P s ) + n E(P s )] + n ( p + n p) Pooled estimators Mean and Variance needed for Ci and hypothesis testing Size Mean Variance Sample I X s Sample II n X s Given in formula booklet S p Mean m = X + n X + n Variance Using sample variances s = n s + n s + n Using unbiased Estimators of Population Variances Sample Variance (σ n ) on calculator ( )S + (n )S + n Using summary values s = S (x i x ) + S (x j x j ) + n
4 3. Confidence Intervals Interpretation of a 95% CI different samples of size n lead to different values of the estimator and hence to different 95% confidence Intervals. On average 95% of these intervals will contain the true population value. Difference between means Assumptions o A Normal Distribution is stated or can be assumed o Unknown Population Variance o Small samples are used x x ± t c s + n t c ttables n degrees of freedom 95% look up s = ( )S + (n )S + n If the confidence interval includes 0 we can say that we are 95% confident that there is no difference between the means of the two populations. Population Variance (or Standard Deviation)  Uses s unbiased estimate of population variance (s n )  Uses chisquared c L (lower) c U (upper) so for 95% use 0.5 and n  degrees of freedom (n )s c U < s < (n )s c L Standard Deviation Confidence Interval Work as for variance but square root the final answers Ratio of two normal population Variances  uses s unbiased estimate of population variance (s n )  uses FDistribution must get the degrees of freedom in the correct order Sample X Sample size = n x Degrees of freedom v x = n x  Sample Y Sample size = n y Degrees of freedom v y = n y  If looking for 90% Confidence Interval use p = 0.95 (5% at upper and lower but use the upper limit to find the values of F) numerator denominator F = v F vy x F = F v y v x s x sx F s y s y F If the confidence interval includes it is reasonable to conclude that the two population variances are e qual.
5 4. Hypothesis Testing For each type of test state Null hypothesis H 0 : m X = m Y Alternative Hypothesis H : m x > m y ( tail test) State significance level and distribution Determine critical value/ region sketch graph Calculate the appropriate test statistic Conclude accept or reject H o in favour of H MEANS s is the pooled sample variance Difference between means Two small samples and n have mean values x and x Test statistic t = x x s + n Distribution Use t tables +n degrees of freedom H : m ¹ m x y H : m < m x y H : m > m x y tailed test tailed test tailed test Assumptions  the two populations are Normal  the two populations have the same variance Remember to divide your rejection region critical value Significance level by or critical region Difference between matches pairs Paired Samples If samples can be paired exactly, the difference between the pairs of values can be tested to see if they form a distribution with zero mean, assumed to be normal As we don t known the population variances of these differences  use the tdistribution Test statistic t = d 0 d is the mean of the differences of the matches pairs s is the unbiased estimate of the variance of the differences of s the matches pairs n Distribution t distribution with n  degrees of freedom ( n= number of pairs used)
6 VARIANCES (Standard Deviations always work in terms of variance) Tests about a SINGLE population variance tail test tailed test Test Statistic Distribution H 0 : s =s 0 H 0 : s = s 0 Chisquared H : s > s 0 H : s (n )s s 0 c = s n degrees of freedom 0 H : s < s 0 Assumption population is approximately normal Comparison of population variances  can be used to check that the variances are roughly the same (one of the assumptions needed to use the tdistribution when comparing means)  uses the ratio of the two population variances compares to  always have the larger variance as the numerator ONE TAILED TESTS (rare in an exam) NUMERATOR n  degrees of freedom H : s > s or H : s > s Rejection Region F > F F = s V Test statistic s F = V Test statistic s TWO TAILED TESTS H : s s F = s s if s > s s or F = s if s > s s DENOMINATOR n  degrees of freedom Rejection Region  as above but remember to divide the significance level by A scientist records lengths of worms in fields A and B Field A (cm) Field B (cm) Assuming that these are random samples from normal populations, test at the 5% significance level that the population variances are equal. H 0 : s A = s B H : s A s B F distribution tailed test in tables S A = n A = 6 Degrees of freedom = 5 S B =.545 n B = 5 Degrees of freedom = Test Statistic F = = 3.5 F 54 = 7.39 As 3.5 < 7.39 no significant evidence at the 5% level to indicate that the variances are not equal : Accept H 0
7 5. Goodness of Fit ChiSquared n X = S i = (O i E i ) E i Expected Frequencies must be 5 Degrees of freedom : if there are k groups (in your X calculation) and p parameters are estimated, then the no. of degrees of freedom is kp. The observed X is compared with c onesided tables. If X is too high, we reject the hypothesis that this is the correct model for the distribution. Formula book contains the functions for the Binomial use n  degrees of freedom if you have estimated p from the data Poisson use n degrees of freedom if you have estimated λ from the data you may need to make the last group k Geometric you may need to make the last group k Uniform also test for independence e.g. if number customers is independent of the day of the week then each day would have the same frequency Normal standardise and use tables to find the probabilities z = x m s n 3 degrees of freedom if mean and variance estimated from data use and at the lower and upper limits to ensure all covered Analysis of the goals scored per match by a football team gave the following results. Goals per match (x) Matches (f) Test at the 5% level whether the distribution can be modelled by a Poisson distribution. ALWAYS start with a hypothesis H 0 : The distribution is Poisson Significance Level 5% From the data mean λ =.3 P(X = x) = e 3 (3) x Observed Expected x! Extra group added X = 4.75 compare to c (5%) with 4 degrees of freedom c = 9.49 As X < 9.49 we do not reject H 0 and conclude that the distribution follows a Poisson Distribution having the same mean.
This paper is not to be removed from the Examination Halls
~~ST104B ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON ST104B ZB BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
More informationTopic 3: The Expectation of a Random Variable
Topic 3: The Expectation of a Random Variable Course 003, 2017 Page 0 Expectation of a discrete random variable Definition (Expectation of a discrete r.v.): The expected value (also called the expectation
More informationChapter 4. Continuous Random Variables
Chapter 4. Continuous Random Variables Review Continuous random variable: A random variable that can take any value on an interval of R. Distribution: A density function f : R R + such that 1. nonnegative,
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationAdvanced Herd Management Probabilities and distributions
Advanced Herd Management Probabilities and distributions Anders Ringgaard Kristensen Slide 1 Outline Probabilities Conditional probabilities Bayes theorem Distributions Discrete Continuous Distribution
More informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566  Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
More informationAnalysis of Engineering and Scientific Data. Semester
Analysis of Engineering and Scientific Data Semester 1 2019 Sabrina Streipert s.streipert@uq.edu.au Example: Draw a random number from the interval of real numbers [1, 3]. Let X represent the number. Each
More informationCSE 312 Final Review: Section AA
CSE 312 TAs December 8, 2011 General Information General Information Comprehensive Midterm General Information Comprehensive Midterm Heavily weighted toward material after the midterm PreMidterm Material
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 informationThis does not cover everything on the final. Look at the posted practice problems for other topics.
Class 7: Review Problems for Final Exam 8.5 Spring 7 This does not cover everything on the final. Look at the posted practice problems for other topics. To save time in class: set up, but do not carry
More informationProbability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
More informationS n = x + X 1 + X X n.
0 Lecture 0 0. Gambler Ruin Problem Let X be a payoff if a coin toss game such that P(X = ) = P(X = ) = /2. Suppose you start with x dollars and play the game n times. Let X,X 2,...,X n be payoffs in each
More informationCONTINUOUS RANDOM VARIABLES
the Further Mathematics network www.fmnetwork.org.uk V 07 REVISION SHEET STATISTICS (AQA) CONTINUOUS RANDOM VARIABLES The main ideas are: Properties of Continuous Random Variables Mean, Median and Mode
More informationMock Exam  2 hours  use of basic (nonprogrammable) calculator is allowed  all exercises carry the same marks  exam is strictly individual
Mock Exam  2 hours  use of basic (nonprogrammable) calculator is allowed  all exercises carry the same marks  exam is strictly individual Question 1. Suppose you want to estimate the percentage of
More informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 10: Expectation and Variance Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/ psarkar/teaching
More informationMATH 3510: PROBABILITY AND STATS June 15, 2011 MIDTERM EXAM
MATH 3510: PROBABILITY AND STATS June 15, 2011 MIDTERM EXAM YOUR NAME: KEY: Answers in Blue Show all your work. Answers out of the blue and without any supporting work may receive no credit even if they
More informationMark Scheme (Results) June 2008
Mark (Results) June 8 GCE GCE Mathematics (6684/) Edexcel Limited. Registered in England and Wales No. 44967 June 8 6684 Statistics S Mark Question (a) (b) E(X) = Var(X) = ( ) x or attempt to use dx µ
More informationSTATISTICS 1 REVISION NOTES
STATISTICS 1 REVISION NOTES Statistical Model Representing and summarising Sample Data Key words: Quantitative Data This is data in NUMERICAL FORM such as shoe size, height etc. Qualitative Data This is
More informationIB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice
IB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice 1. We have seen that the TI83 calculator random number generator X = rand defines a uniformlydistributed random
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 information2.3 Analysis of Categorical Data
90 CHAPTER 2. ESTIMATION AND HYPOTHESIS TESTING 2.3 Analysis of Categorical Data 2.3.1 The Multinomial Probability Distribution A mulinomial random variable is a generalization of the binomial rv. It results
More informationCounting principles, including permutations and combinations.
1 Counting principles, including permutations and combinations. The binomial theorem: expansion of a + b n, n ε N. THE PRODUCT RULE If there are m different ways of performing an operation and for each
More informationPage Max. Possible Points Total 100
Math 3215 Exam 2 Summer 2014 Instructor: Sal Barone Name: GT username: 1. No books or notes are allowed. 2. You may use ONLY NONGRAPHING and NONPROGRAMABLE scientific calculators. All other electronic
More informationEstimating the accuracy of a hypothesis Setting. Assume a binary classification setting
Estimating the accuracy of a hypothesis Setting Assume a binary classification setting Assume input/output pairs (x, y) are sampled from an unknown probability distribution D = p(x, y) Train a binary classifier
More informationMA/ST 810 MathematicalStatistical Modeling and Analysis of Complex Systems
MA/ST 810 MathematicalStatistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationThings to remember when learning probability distributions:
SPECIAL DISTRIBUTIONS Some distributions are special because they are useful They include: Poisson, exponential, Normal (Gaussian), Gamma, geometric, negative binomial, Binomial and hypergeometric distributions
More informationTUTORIAL 8 SOLUTIONS #
TUTORIAL 8 SOLUTIONS #9.11.21 Suppose that a single observation X is taken from a uniform density on [0,θ], and consider testing H 0 : θ = 1 versus H 1 : θ =2. (a) Find a test that has significance level
More informationStatistics. Statistics
The main aims of statistics 1 1 Choosing a model 2 Estimating its parameter(s) 1 point estimates 2 interval estimates 3 Testing hypotheses Distributions used in statistics: χ 2 ndistribution 2 Let X 1,
More informationClosed book and notes. 60 minutes. Cover page and four pages of exam. No calculators.
IE 230 Seat # Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators. Score Exam #3a, Spring 2002 Schmeiser Closed book and notes. 60 minutes. 1. True or false. (for each,
More informationMAS113 Introduction to Probability and Statistics. Proofs of theorems
MAS113 Introduction to Probability and Statistics Proofs of theorems Theorem 1 De Morgan s Laws) See MAS110 Theorem 2 M1 By definition, B and A \ B are disjoint, and their union is A So, because m is a
More informationSociology 6Z03 Review II
Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability
More informationMath Review Sheet, Fall 2008
1 Descriptive Statistics Math 30705 Review Sheet, Fall 2008 First we need to know about the relationship among Population Samples Objects The distribution of the population can be given in one of the
More information3 Multiple Discrete Random Variables
3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationZIMBABWE SCHOOL EXAMINATIONS COUNCIL (ZIMSEC) ADVANCED LEVEL SYLLABUS
ZIMBABWE SCHOOL EXAMINATIONS COUNCIL (ZIMSEC) ADVANCED LEVEL SYLLABUS Further Mathematics 9187 EXAMINATION SYLLABUS FOR 20132017 2 CONTENTS Page Aims.. 2 Assessment Objective.. 2 Scheme of Assessment
More informationCHAPTER 3 Describing Relationships
CHAPTER 3 Describing Relationships 3.1 Scatterplots and Correlation The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Scatterplots and Correlation Learning
More informationRandom Variables. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 13 Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2013 2 / 13 Random Variable Definition A realvalued
More informationAn inferential procedure to use sample data to understand a population Procedures
Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (zscores), statistics, conclusion Two types of errors
More informationHypothesis Tests and Estimation for Population Variances. Copyright 2014 Pearson Education, Inc.
Hypothesis Tests and Estimation for Population Variances 111 Learning Outcomes Outcome 1. Formulate and carry out hypothesis tests for a single population variance. Outcome 2. Develop and interpret confidence
More informationWeek 2: Review of probability and statistics
Week 2: Review of probability and statistics Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2017 c 2017 PERRAILLON ALL RIGHTS RESERVED
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationBrandon C. Kelly (Harvard Smithsonian Center for Astrophysics)
Brandon C. Kelly (Harvard Smithsonian Center for Astrophysics) Probability quantifies randomness and uncertainty How do I estimate the normalization and logarithmic slope of a X ray continuum, assuming
More informationSolutions to the Spring 2015 CAS Exam ST
Solutions to the Spring 2015 CAS Exam ST (updated to include the CAS Final Answer Key of July 15) There were 25 questions in total, of equal value, on this 2.5 hour exam. There was a 10 minute reading
More informationSmoking Habits. Moderate Smokers Heavy Smokers Total. Hypertension No Hypertension Total
Math 3070. Treibergs Final Exam Name: December 7, 00. In an experiment to see how hypertension is related to smoking habits, the following data was taken on individuals. Test the hypothesis that the proportions
More informationMAS113 Introduction to Probability and Statistics. Proofs of theorems
MAS113 Introduction to Probability and Statistics Proofs of theorems Theorem 1 De Morgan s Laws) See MAS110 Theorem 2 M1 By definition, B and A \ B are disjoint, and their union is A So, because m is a
More informationEE 345 MIDTERM 2 Fall 2018 (Time: 1 hour 15 minutes) Total of 100 points
Problem (8 points) Name EE 345 MIDTERM Fall 8 (Time: hour 5 minutes) Total of points How many ways can you select three cards form a group of seven nonidentical cards? n 7 7! 7! 765 75 = = = = = = 35 k
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 informationf (1 0.5)/n Z =
Math 466/566  Homework 4. We want to test a hypothesis involving a population proportion. The unknown population proportion is p. The null hypothesis is p = / and the alternative hypothesis is p > /.
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More informationPreliminary Statistics. Lecture 3: Probability Models and Distributions
Preliminary Statistics Lecture 3: Probability Models and Distributions Rory Macqueen (rm43@soas.ac.uk), September 2015 Outline Revision of Lecture 2 Probability Density Functions Cumulative Distribution
More information1 Review of Probability and Distributions
Random variables. A numerically valued function X of an outcome ω from a sample space Ω X : Ω R : ω X(ω) is called a random variable (r.v.), and usually determined by an experiment. We conventionally denote
More informationTopic 3: The Expectation of a Random Variable
Topic 3: The Expectation of a Random Variable Course 003, 2016 Page 0 Expectation of a discrete random variable Definition: The expected value of a discrete random variable exists, and is defined by EX
More informationEvaluating Hypotheses
Evaluating Hypotheses IEEE Expert, October 1996 1 Evaluating Hypotheses Sample error, true error Confidence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution,
More informationCopyright c 2006 Jason Underdown Some rights reserved. choose notation. n distinct items divided into r distinct groups.
Copyright & License Copyright c 2006 Jason Underdown Some rights reserved. choose notation binomial theorem n distinct items divided into r distinct groups Axioms Proposition axioms of probability probability
More informationQ Scheme Marks AOs. Notes. Ignore any extra columns with 0 probability. Otherwise 1 for each. If 4, 5 or 6 missing B0B0.
1a k(16 9) + k(25 9) + k(36 9) (or 7k + 16k + 27k). M1 2.1 4th = 1 M1 Þ k = 1 50 (answer given). * Model simple random variables as probability (3) 1b x 4 5 6 P(X = x) 7 50 16 50 27 50 Note: decimal values
More informationexp{ (x i) 2 i=1 n i=1 (x i a) 2 (x i ) 2 = exp{ i=1 n i=1 n 2ax i a 2 i=1
4 Hypothesis testing 4. Simple hypotheses A computer tries to distinguish between two sources of signals. Both sources emit independent signals with normally distributed intensity, the signals of the first
More informationMark Scheme (Results) Summer 2009
Mark (Results) Summer 009 GCE GCE Mathematics (6684/01) June 009 6684 Statistics S Mark Q1 [ X ~ B(0,0.15) ] P(X 6), = 0.8474 awrt 0.847 Y ~ B(60,0.15) Po(9) for using Po(9) P(Y < 1), = 0.8758 awrt 0.876
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationECE 313 Probability with Engineering Applications Fall 2000
Exponential random variables Exponential random variables arise in studies of waiting times, service times, etc X is called an exponential random variable with parameter λ if its pdf is given by f(u) =
More informationLectures on Statistics. William G. Faris
Lectures on Statistics William G. Faris December 1, 2003 ii Contents 1 Expectation 1 1.1 Random variables and expectation................. 1 1.2 The sample mean........................... 3 1.3 The sample
More informationPROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION DEFINITION: If S is a sample space with a probability measure and x is a real valued function defined over the elements of S, then x is called a random variable. Types of Random
More informationChapter 23: Inferences About Means
Chapter 3: Inferences About Means Sample of Means: number of observations in one sample the population mean (theoretical mean) sample mean (observed mean) is the theoretical standard deviation of the population
More informationCommon Discrete Distributions
Common Discrete Distributions Statistics 104 Autumn 2004 Taken from Statistics 110 Lecture Notes Copyright c 2004 by Mark E. Irwin Common Discrete Distributions There are a wide range of popular discrete
More informationHomework 4 Solution, due July 23
Homework 4 Solution, due July 23 Random Variables Problem 1. Let X be the random number on a die: from 1 to. (i) What is the distribution of X? (ii) Calculate EX. (iii) Calculate EX 2. (iv) Calculate Var
More informationPractice Problems Section Problems
Practice Problems Section 443 44 45 46 47 48 410 Supplemental Problems 41 to 49 413, 14, 15, 17, 19, 0 43, 34, 36, 38 447, 49, 5, 54, 55 459, 60, 63 466, 68, 69, 70, 74 479, 81, 84 485,
More informationThe ttest: A zscore for a sample mean tells us where in the distribution the particular mean lies
The ttest: So Far: Sampling distribution benefit is that even if the original population is not normal, a sampling distribution based on this population will be normal (for sample size > 30). Benefit
More informationDistributions of Functions of Random Variables. 5.1 Functions of One Random Variable
Distributions of Functions of Random Variables 5.1 Functions of One Random Variable 5.2 Transformations of Two Random Variables 5.3 Several Random Variables 5.4 The MomentGenerating Function Technique
More informationSemester , Example Exam 1
Semester 1 2017, Example Exam 1 1 of 10 Instructions The exam consists of 4 questions, 14. Each question has four items, ad. Within each question: Item (a) carries a weight of 8 marks. Item (b) carries
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 informationBINOMIAL DISTRIBUTION
BINOMIAL DISTRIBUTION The binomial distribution is a particular type of discrete pmf. It describes random variables which satisfy the following conditions: 1 You perform n identical experiments (called
More informationFundamental Tools  Probability Theory II
Fundamental Tools  Probability Theory II MSc Financial Mathematics The University of Warwick September 29, 2015 MSc Financial Mathematics Fundamental Tools  Probability Theory II 1 / 22 Measurable random
More informationab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More information(It's not always good, but we can always make it.) (4) Convert the normal distribution N to the standard normal distribution Z. Specically.
. Introduction The quick summary, going forwards: Start with random variable X. 2 Compute the mean EX and variance 2 = varx. 3 Approximate X by the normal distribution N with mean µ = EX and standard deviation.
More informationStochastic Models of Manufacturing Systems
Stochastic Models of Manufacturing Systems Ivo Adan Organization 2/47 7 lectures (lecture of May 12 is canceled) Studyguide available (with notes, slides, assignments, references), see http://www.win.tue.nl/
More information(b). What is an expression for the exact value of P(X = 4)? 2. (a). Suppose that the moment generating function for X is M (t) = 2et +1 3
Math 511 Exam #2 Show All Work 1. A package of 200 seeds contains 40 that are defective and will not grow (the rest are fine). Suppose that you choose a sample of 10 seeds from the box without replacement.
More informationPart IA Probability. Theorems. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Theorems Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationStatistics 224 Solution key to EXAM 2 FALL 2007 Friday 11/2/07 Professor Michael Iltis (Lecture 2)
NOTE : For the purpose of review, I have added some additional parts not found on the original exam. These parts are indicated with a ** beside them Statistics 224 Solution key to EXAM 2 FALL 2007 Friday
More informationSome Continuous Probability Distributions: Part I. Continuous Uniform distribution Normal Distribution. Exponential Distribution
Some Continuous Probability Distributions: Part I Continuous Uniform distribution Normal Distribution Exponential Distribution 1 Chapter 6: Some Continuous Probability Distributions: 6.1 Continuous Uniform
More informationStatistical distributions: Synopsis
Statistical distributions: Synopsis Basics of Distributions Special Distributions: Binomial, Exponential, Poisson, Gamma, ChiSquare, F, Extremevalue etc Uniform Distribution Empirical Distributions Quantile
More informationCS 5014: Research Methods in Computer Science. Bernoulli Distribution. Binomial Distribution. Poisson Distribution. Clifford A. Shaffer.
Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 2015 by Clifford A. Shaffer Computer Science Title page Computer Science Clifford A. Shaffer Fall 2015 Clifford A. Shaffer
More informationEXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS
EXAM Exam # Math 3342 Summer II, 2 July 2, 2 ANSWERS i pts. Problem. Consider the following data: 7, 8, 9, 2,, 7, 2, 3. Find the first quartile, the median, and the third quartile. Make a box and whisker
More informationMark Scheme (Results) Summer 2007
Mark Scheme (Results) Summer 007 GCE GCE Mathematics Statistics S3 (6691) Edexcel Limited. Registered in England and Wales No. 4496750 Registered Office: One90 High Holborn, London WC1V 7BH June 007 6691
More informationChapter 5. Means and Variances
1 Chapter 5 Means and Variances Our discussion of probability has taken us from a simple classical view of counting successes relative to total outcomes and has brought us to the idea of a probability
More informationPHP2510: Principles of Biostatistics & Data Analysis. Lecture X: Hypothesis testing. PHP 2510 Lec 10: Hypothesis testing 1
PHP2510: Principles of Biostatistics & Data Analysis Lecture X: Hypothesis testing PHP 2510 Lec 10: Hypothesis testing 1 In previous lectures we have encountered problems of estimating an unknown population
More information# of 6s # of times Test the null hypthesis that the dice are fair at α =.01 significance
Practice Final Exam Statistical Methods and Models  Math 410, Fall 2011 December 4, 2011 You may use a calculator, and you may bring in one sheet (8.5 by 11 or A4) of notes. Otherwise closed book. The
More informationDisjointness and Additivity
Midterm 2: Format Midterm 2 Review CS70 Summer 2016  Lecture 6D David Dinh 28 July 2016 UC Berkeley 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one doublesided sheet) of handwritten
More informationMidterm 2 Review. CS70 Summer Lecture 6D. David Dinh 28 July UC Berkeley
Midterm 2 Review CS70 Summer 2016  Lecture 6D David Dinh 28 July 2016 UC Berkeley Midterm 2: Format 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one doublesided sheet) of handwritten
More informationBasic Probability Reference Sheet
February 27, 2001 Basic Probability Reference Sheet 17.846, 2001 This is intended to be used in addition to, not as a substitute for, a textbook. X is a random variable. This means that X is a variable
More informationSTAT 430/510: Lecture 16
STAT 430/510: Lecture 16 James Piette June 24, 2010 Updates HW4 is up on my website. It is due next Mon. (June 28th). Starting today back at section 6.7 and will begin Ch. 7. Joint Distribution of Functions
More informationCME 106: Review Probability theory
: Probability theory Sven Schmit April 3, 2015 1 Overview In the first half of the course, we covered topics from probability theory. The difference between statistics and probability theory is the following:
More informationECE 302 Division 1 MWF 10:3011:20 (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 information, 0 x < 2. a. Find the probability that the text is checked out for more than half an hour but less than an hour. = (1/2)2
Math 205 Spring 206 Dr. Lily Yen Midterm 2 Show all your work Name: 8 Problem : The library at Capilano University has a copy of Math 205 text on twohour reserve. Let X denote the amount of time the text
More informationReading Material for Students
Reading Material for Students Arnab Adhikari Indian Institute of Management Calcutta, Joka, Kolkata 714, India, arnaba1@email.iimcal.ac.in Indranil Biswas Indian Institute of Management Lucknow, Prabandh
More information2. Topic: Series (Mathematical Induction, Method of Difference) (i) Let P n be the statement. Whenn = 1,
GCE A Level October/November 200 Suggested Solutions Mathematics H (9740/02) version 2. MATHEMATICS (H2) Paper 2 Suggested Solutions 9740/02 October/November 200. Topic:Complex Numbers (Complex Roots of
More informationTable of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).
Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X.04) =.8508. For z < 0 subtract the value from,
More informationChapter 24. Comparing Means
Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing
More informationREVIEW: Midterm Exam. Spring 2012
REVIEW: Midterm Exam Spring 2012 Introduction Important Definitions:  Data  Statistics  A Population  A census  A sample Types of Data Parameter (Describing a characteristic of the Population) Statistic
More informationTest Problems for Probability Theory ,
1 Test Problems for Probability Theory 010616, 010114 1. Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 30
More informationGuidelines for Solving Probability Problems
Guidelines for Solving Probability Problems CS 1538: Introduction to Simulation 1 Steps for Problem Solving Suggested steps for approaching a problem: 1. Identify the distribution What distribution does
More information