Inter-Annual Variability and Uncertainty in Wind Farm Annual Energy Production Estimates
|
|
- Martina Reeves
- 6 years ago
- Views:
Transcription
1 Inter-Annual Variability and Uncertainty in Wind Farm Annual Energy Production Estimates Peter Taylor 1,2, Jim Salmon 2, 1 York University, 2 Zephyr North Canada Contact: pat@yorku.ca ICEM, Toulouse, June 2013
2 Uncertainties in the estimated Annual Energy Production (AEP) are crucial factors in assessing the financial viability of a wind farm project. A good estimate of the mean (P50) value is the starting point but finance agencies give at least as much weight to the P90, P95 or P99 estimates of future production, both annual and averaged over multiple years. These are based on uncertainty estimations and the Inter- Annual Variability (IAV) of the wind resource plays an important role. When acquisition of a portfolio of spatially distributed wind farms is under consideration then correlations between the wind speeds at the different farm locations play a role in assessing the overall IAV for the portfolio.
3 A structure is presented for uncertainties in the estimation of annual energy production for wind farms. The typical industry practice of assuming 6% variability in wind IAV is too conservative for many locations. IAV at hub height appears to be lower than the IAV at 10m. At hub height a value of 3% appears to be appropriate but 4% may represent a conservative compromise for most North American locations. For investment in a portfolio of multiple wind farm projects, a less than 100% correlation between winds at different sites will reduce the uncertainties in AEP. A structure for assessing the benefits of the "portfolio effect" is presented and some sample Canadian cases discussed.
4 Fixed Uncertainties a) Initial AEP estimate errors - wind measurement, and flow and wake modelling. Hardware uncertainties (e.g. power curves). Uncertainties that can vary Year to Year b) Operational uncertainties (turbine failures, maintenance, grid connection availability etc.) c) Meteorological factors, icing and inter-annual variability (IAV). Uncertainty is central in the prediction of the secure (P90) multi-year return on investments.
5 Single Wind Farm We denote the long term AEP (P50) by P50. Perturbations from P50 in any given year are denoted by p' with a long term mean of zero. If the p' are normally distributed and if we denote the root mean square of p' over many years as σ, then the 1-year P90 is P(90-1 year) = P σ (1-year) The same relationship would hold for multi-year P90 (Y-year) values with a reduced multi-year standard deviation (assuming each year is independent) as, σ (Y-year) = σ (1-year)/ Y But not all uncertainties will vary from year to year - e.g. the fixed errors such as possible errors in the resource assessment, so...
6 Denoting uncertainties in AEP as u = f + g where the f are fixed uncertainties associated with pre-construction estimates and g are uncertainties which will vary from year to year with zero long term mean, the multi-year uncertainties are assumed to be given as u (Y )= f 2 +g 2 /Y where f and g are not correlated. As an example, consider a case with yearly wind IAV and random technical variability giving a relative production uncertainty (g/p50) of 8% and the relative fixed uncertainties (f/p50), associated with original measurement errors, modelling errors, turbine variability, etc., of 8%. Combined, these give a 1-year relative AEP uncertainty (square root of sum of squares) of 11.3% and a 10- year average AEP uncertainty of 8.4% mostly associated with the potential errors in the pre-construction wind monitoring, MCP analysis, wind farm modelling and turbine performance. IAV impacts will be smoothed out by averaging over 10 years but the fixed uncertainty remains at 8%.
7 Inter-Annual Variability (IAV) Many groups assume wind speed IAV = 6%, based in part on an analysis of 22+ years of annual wind speeds at 11 UK met stations. (6.2% IAV) Raftery A., Tindal J. and Garrad A. (1997), Understanding the risks of financing windfarms, Proc. EWEA Wind Energy Conference, Dublin. and on the EWEA web site, or Fig in the 2009 EWEA book Wind Energy - The Facts ( %)
8 EWEA/GLGH: Wind Map of Europe Inter annual variations. Shown as Standard Deviation as a percentage of mean
9 Inter-Annual Variability (IAV) Effects of trend. Suppose X(t) = X 0 + at + x' for -T/2 < t < T/2 where <x'> = 0, <x' 2 > = s T 2 and averages <..> are over the interval from -T/2 to T/2. Then <X(t)> = X 0, but <(X(t) - X 0 ) 2 > = <(at) 2 > + 2a <tx'> + <x' 2 > We can note that <(at) 2 > = a 2 T 2 /12,for the continuous case. Assuming <tx'> = 0, and using the continuous case result, the IAV computed would be σ T 2 = (s T 2 + a 2 T 2 /12) 1/2
10 Results from Cabauw Values computed from annual mean winds at the Cabauw tower 100m (interpolated) 10m Year Range Mean σ a s s/mean Mean σ a s s/mean N Table 1 Inter-Annual Variability computed from yearly-averaged data from the Cabauw tower. Wind speeds and standard deviations are in m/s. σ is the standard deviation computed directly from the annual averages over the year range indicated. A linear fit to the data gives slopes, a (m/s/year) and s is the standard deviation after trend removal. N is the number of annual data values used. Note that no data were available Data kindly provided by Fred Bosveld (KNMI). Note reductions in s/mean at 100m versus 10m.
11 Inter-Annual Variability (IAV) Length of record: If there is no trend, or trend has been ignored, and if individual annual averages are considered as independent and identically distributed (IID) with a normal distribution then an average over T values would have a variance of σ2/t, where σ is the standard deviation of the complete data set of IID annual averages. If this is regarded as an error relative to the true mean then the computed variance from the limited data set (σt 2 ) would, on average, be reduced relative to the variance (σ2) relative to the long term mean by a factor (1-1/T) since, with sums over i = 1 to T, σ 2 = (1/T) (U i - U ) 2 = (1/T) (U i - U T + U T - U ) 2 = (1/T) {( U i - U T ) 2 + (U T - U ) 2 } since (U i - U T ) = 0
12 If we average over many cases, σ 2 = σ T 2 + σ 2 /T, This implies that, on average, σ = σ T (1-1/T) -1/2 So for an 11 year sample, assuming no trend, or after trend removal, we could increase the computed IAV by a factor For Cabauw 100m data we would have long term IAV estimates of 3.6 and 4.0% over the two 11-year periods for which data are available.
13 IAV at Canadian Surface Stations MSC station wind Speeds m/s Broadview, Weyburn, Regina Indian Hd, Egbert Location Mean SK 4.17 SK 4.91 Apt, SK 5.22 SK 4.85 CS, ON 3.23 σ T σ T /Mean Hamilton, Goderich, Windsor, Wiarton, Mt Forest, Location Mean ON 4.21 ON 4.70 ON 4.27 ON 3.62 ON 3.06 σ T σ T /Mean
14 Wind Speed (m/s) Broadview, SK Weyburn, SK Regina Airport, SK Indian Head, SK Egbert, ON Hamilton Airport, ON Goderich, ON Windsor Airport, ON Wiarton, ON Mount Forest, ON Average (8 stations) Fit 1: Linear Calendar Year
15 IAV in reanalysis data for Canadian locations No Canadian long term tall tower data were available. Alternative sources of wind information are provided by the "re-analysis" data sets from the National Centre for Environmental Prediction (NCEP) or the European Centre for Medium-Range Weather Forecasting (ECMWF). At least two commercial organizations can supply down-scaled, site-specific data sets for potential wind farm sites. The analysis to be presented here uses hourly SERIES data from the Vortex company (Vortex Factoria de Calculs S.L. located in Barcelona, Spain). We use their results based on down-scaling to 3 km resolution using site-centred Weather Research and Forecasting Model (WRF) simulations driven by NCAR-NCEP CFSR reanalysis data. The down-scaled site-specific data cover eleven complete years from January 1, 2001 to Dec 31, 2011, and can be interpolated to hub heights which we will take as 100m.
16 Typical comparison between hourly VORTEX series and tower data. We find that they generally track well but there may be a scaling factor and some slight seasonal variation - which would not significantly impact IAV.
17 1.08 Normalized annual wind speed BC1 AB1 SK 1 SK2 SK3 ON1 ON3 ON5 Average Linear Fit Year Annual Average 100m winds, Vortex SERIES Vortex SERIES annual average wind speeds at 8 Canadian locations. Note expanded scale.
18 Note Standard Deviations = IAV Normalized speeds Location BC1 AB1 SK1 SK2 SK3 ON4 ON1 ON3 ON5 ON2 Hub Height 100m 100m 100m 100m 100m 100m 100m 100m 100m 100m Year Mean Std Dev So IAV < 3% except for BC1 (3.7%)
19 Portfolios If we had production data for N wind farms with individual long term average productions Pi and perturbations in any given year p'i, we would let The long term variance of the total portfolio perturbation could be written as, N p ' 2 =[( i=1 p ' i ) 2 N ]= i=1 N P= i =1 N [ p ' 2 i ]+2 i=1 where [... ] indicates a long term average. In the simple case where all <p'i 2 > are equal to σ 2, the overall standard deviation, p', could vary from N 1/2 σ with no correlation to Nσ if there is 100% correlation. P i N [ p ' i p ' j ] j=i+1
20 Portfolios In terms of estimated uncertainties in AEP, let the uncertainty for each wind farm be u i = f i + w i + t i where the f i are fixed uncertainties, w i are AEP uncertainties caused by the wind IAV and t i are uncertainties expected to vary from year to year associated with technical and similar problems.[... ] indicates a long term average. Then the total estimated, annual AEP uncertainty for the portfolio of N wind farms becomes, N [( i=1 u i ) 2 N ]= i=1 N ([ f 2 i ]+[w 2 i ]+[t 2 i ])+2 i=1 N [w i w j ] j=i+1 assuming independence of all except the wind variability. So we need to look at wind induced AEP uncertainties and their correlations.
21 A few portfolio results To illustrate the correlations we have computed gross AEP for six imaginary wind farm projects at locations across Canada based on Vortex SERIES wind data. The same idealized 2 MW, 100 m hub height turbine is used at each location and each farm is assumed to have the same number of turbines. The locations are organized from West (BC1) to East (ON5). The correlations decrease with increasing separation with a slightly negative correlation between ON5 and BC1. For a portfolio of all 6 wind farms the (normalized) double sum N N < i= 1 j= i+ 1 w i w j 6.34 can be compared to a sum of 15 (from (N-i) summed from 1 to N) if there were correlations of 1.0 between each farm. > =
22 AEP cross-correlations CROSS CORRELATIONS Location BC1 AB1 SK1 ON1 ON3 ON5 BC AB SK ON ON ON Cross-correlations of AEP perturbations from six imaginary wind farm projects across Canada. All calculations based on Vortex SERIES hourly wind data at locations indicated with a single,representative wind turbine and a generic 2MW power curve. Locations are from West to East. Orange and Yellow highlights indicate smaller portfolios.
23 Conclusions A structure is presented for uncertainties in the estimation of annual energy production for wind farms. Inter-annual variability in the average hub-height wind speed plays a significant part in the uncertainty and it is argued that the typical industry practice of assuming 6% variability is too conservative for many locations. IAV at hub height appears to be less than at 10 m. A hub height value of 3% may be more realistic - at least for Canada. Site specific estimation based on reanalysis data is also a viable method. For investment in a portfolio of multiple wind farm projects, a less than 100% correlation between winds at different sites will reduce the uncertainties in total AEP. A structure for assessing the benefits of the "portfolio effect" is presented and some sample cases discussed. Acknowledgements We are grateful to a number of Zephyr North clients for permission to make use of their Vortex SERIES data for part of this research.
A PRESENTATION BY THE AMERICAN ACADEMY OF ACTUARIES TO THE NAIC S CLIMATE CHANGE AND GLOBAL WARMING (C) WORKING GROUP
A PRESENTATION BY THE AMERICAN ACADEMY OF ACTUARIES TO THE NAIC S CLIMATE CHANGE AND GLOBAL WARMING (C) WORKING GROUP MARCH 24, 2018 MILWAUKEE, WISCONSIN COPYRIGHT 2018 2018 American Academy of Actuaries.
More informationStat3011: Solution of Midterm Exam One
1 Stat3011: Solution of Midterm Exam One Fall/2003, Tiefeng Jiang Name: Problem 1 (30 points). Choose one appropriate answer in each of the following questions. 1. (B ) The mean age of five people in a
More informationChapter 7: Random Variables
Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value
More informationEnergy Yield Reconciliation in Monthly O&M Reports
Energy Yield Reconciliation in Monthly O&M Reports Claire Puttock, Lee Cameron & Alex Clerc April 15, 2016 EWEA Technology Workshop, Bilbao Contents Introduction to variance explanation Motivation Key
More informationProspects for Wind Farm Installation in Wapakoneta, Ohio: An Initial Study on Economic Feasibility
Prospects for Wind Farm Installation in Wapakoneta, Ohio: An Initial Study on Economic Feasibility Prepared by Katherine Dykes 12/04/2007 ESD 71 Prof. de Neufville Bowling Green, Ohio Wind Farm Content
More informationModelling Returns: the CER and the CAPM
Modelling Returns: the CER and the CAPM Carlo Favero Favero () Modelling Returns: the CER and the CAPM 1 / 20 Econometric Modelling of Financial Returns Financial data are mostly observational data: they
More informationTHE FINANCIAL IMPACT OF WIND PLANT UNCERTAINTY
ALBANY BARCELONA BANGALORE NAWEA Symposium Aug 6, 2013 Boulder, CO THE FINANCIAL IMPACT OF WIND PLANT UNCERTAINTY BRUCE H. BAILEY, PH.D., CCM PRESIDENT/CEO AWS TRUEPOWER, LLC 463 NEW KARNER ROAD ALBANY,
More informationQuality of Wind BMU PNs pp11_47
Quality of Wind BMU PNs pp11_47 Place your chosen image here. The four corners must just cover the arrow tips. For covers, the three pictures should be the same size and in a straight line. by Erol Chartan
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationHow multi-technology PPA structures could help companies reduce risk
How multi-technology PPA structures could help companies reduce risk 1 How multi-technology PPA structures could help companies reduce risk Table of contents Introduction... 3 Key PPA risks related to
More information4.2 Probability Distributions
4.2 Probability Distributions Definition. A random variable is a variable whose value is a numerical outcome of a random phenomenon. The probability distribution of a random variable tells us what the
More informationBloomberg. Portfolio Value-at-Risk. Sridhar Gollamudi & Bryan Weber. September 22, Version 1.0
Portfolio Value-at-Risk Sridhar Gollamudi & Bryan Weber September 22, 2011 Version 1.0 Table of Contents 1 Portfolio Value-at-Risk 2 2 Fundamental Factor Models 3 3 Valuation methodology 5 3.1 Linear factor
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Discrete and Continuous Random
More informationChapter 6 BLM Answers
Chapter 6 BLM Answers BLM 6 2 Chapter 6 Prerequisite Skills 1. a) 0.50, 50% 0.60, 60% 2.3, 233.3% d) 3, 300% 108 km/h 160 m/km 50 m/min 3. 1.99 m 4. a) Time Worked, t (h) Earnings, E ($) 2 30 4 60 6 90
More informationLecture 18 Section Mon, Feb 16, 2009
The s the Lecture 18 Section 5.3.4 Hampden-Sydney College Mon, Feb 16, 2009 Outline The s the 1 2 3 The 4 s 5 the 6 The s the Exercise 5.12, page 333. The five-number summary for the distribution of income
More informationLecture 18 Section Mon, Sep 29, 2008
The s the Lecture 18 Section 5.3.4 Hampden-Sydney College Mon, Sep 29, 2008 Outline The s the 1 2 3 The 4 s 5 the 6 The s the Exercise 5.12, page 333. The five-number summary for the distribution of income
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 7 Estimation: Single Population Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-1 Confidence Intervals Contents of this chapter: Confidence
More informationAP Statistics Chapter 6 - Random Variables
AP Statistics Chapter 6 - Random 6.1 Discrete and Continuous Random Objective: Recognize and define discrete random variables, and construct a probability distribution table and a probability histogram
More informationDiploma in Business Administration Part 2. Quantitative Methods. Examiner s Suggested Answers
Cumulative frequency Diploma in Business Administration Part Quantitative Methods Examiner s Suggested Answers Question 1 Cumulative Frequency Curve 1 9 8 7 6 5 4 3 1 5 1 15 5 3 35 4 45 Weeks 1 (b) x f
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationBasic Procedure for Histograms
Basic Procedure for Histograms 1. Compute the range of observations (min. & max. value) 2. Choose an initial # of classes (most likely based on the range of values, try and find a number of classes that
More informationAdvanced Financial Modeling. Unit 2
Advanced Financial Modeling Unit 2 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation Two Asset Portfolio
More informationRenewing Ireland's Energy European Investment Bank
Renewing Ireland's Energy European Investment Bank IWEA Autumn Conference 2013 Galway Thursday 3 rd October 08/10/2013 1 The European Investment Bank (EIB) Long-term finance promoting European objectives
More informationOvercoming Actuarial Challenges
Overcoming Actuarial Challenges in Crop Insurance August 14, ASI, Mumbai Sonu Agrawal Weather Risk Management Services Ltd Crop Insurance Index Based Assumptive losses based on standard indices Area Yield
More informationChapter 3. Density Curves. Density Curves. Basic Practice of Statistics - 3rd Edition. Chapter 3 1. The Normal Distributions
Chapter 3 The Normal Distributions BPS - 3rd Ed. Chapter 3 1 Example: here is a histogram of vocabulary scores of 947 seventh graders. The smooth curve drawn over the histogram is a mathematical model
More informationOption Pricing Modeling Overview
Option Pricing Modeling Overview Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch) Stochastic time changes Options Markets 1 / 11 What is the purpose of building a
More informationFinal Exam Suggested Solutions
University of Washington Fall 003 Department of Economics Eric Zivot Economics 483 Final Exam Suggested Solutions This is a closed book and closed note exam. However, you are allowed one page of handwritten
More informationAnalysing the effect of the sliding feed-in premium scheme to the revenues of a wind farm with the use of mesoscale data - case study of Greece
Journal of Physics: Conference Series PAPER OPEN ACCESS Analysing the effect of the sliding feed-in premium scheme to the revenues of a wind farm with the use of mesoscale data - case study of Greece To
More informationRandom Variables and Probability Distributions
Chapter 3 Random Variables and Probability Distributions Chapter Three Random Variables and Probability Distributions 3. Introduction An event is defined as the possible outcome of an experiment. In engineering
More informationEnergy Price Processes
Energy Processes Used for Derivatives Pricing & Risk Management In this first of three articles, we will describe the most commonly used process, Geometric Brownian Motion, and in the second and third
More informationCentral Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
More information5.3 Statistics and Their Distributions
Chapter 5 Joint Probability Distributions and Random Samples Instructor: Lingsong Zhang 1 Statistics and Their Distributions 5.3 Statistics and Their Distributions Statistics and Their Distributions Consider
More informationSimultaneous optimization for wind derivatives based on prediction errors
2008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 2008 WeA10.4 Simultaneous optimization for wind derivatives based on prediction errors Yuji Yamada Abstract Wind
More informationC.10 Exercises. Y* =!1 + Yz
C.10 Exercises C.I Suppose Y I, Y,, Y N is a random sample from a population with mean fj. and variance 0'. Rather than using all N observations consider an easy estimator of fj. that uses only the first
More informationThe P99 Hedge That Wasn t
The P99 Hedge That Wasn t An empirical analysis of fixed quantity energy price swap performance for ERCOT wind farms An eye-opening examination of P99 Hedge performance. This should be required reading
More informationMeasures of Dispersion (Range, standard deviation, standard error) Introduction
Measures of Dispersion (Range, standard deviation, standard error) Introduction We have already learnt that frequency distribution table gives a rough idea of the distribution of the variables in a sample
More informationData analysis methods in weather and climate research
Data analysis methods in weather and climate research Dr. David B. Stephenson Department of Meteorology University of Reading www.met.rdg.ac.uk/cag 5. Parameter estimation Fitting probability models he
More informationCandidates Survey February 2010 Q1 - Thinking about the energy sector, how strongly do you agree or disagree which of the following statements?
Table 1/1 Britain should aim to have 15% of its energy from renewable sources by 2020 Agree strongly (2.0) 32 17 10 5 10 9 1 2 3 7 31-1 3 10 9 9 21 11 32% 22% 56% 71% 38% 56% 7% 13% 18% 54% 34% - 17% 27%
More informationLecture 5: Fundamentals of Statistical Analysis and Distributions Derived from Normal Distributions
Lecture 5: Fundamentals of Statistical Analysis and Distributions Derived from Normal Distributions ELE 525: Random Processes in Information Systems Hisashi Kobayashi Department of Electrical Engineering
More informationFinancial Econometrics Review Session Notes 4
Financial Econometrics Review Session Notes 4 February 1, 2011 Contents 1 Historical Volatility 2 2 Exponential Smoothing 3 3 ARCH and GARCH models 5 1 In this review session, we will use the daily S&P
More informationEcon 250 Fall Due at November 16. Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling
Econ 250 Fall 2010 Due at November 16 Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling 1. Suppose a firm wishes to raise funds and there are a large number of independent financial
More informationMean-Variance Analysis
Mean-Variance Analysis Mean-variance analysis 1/ 51 Introduction How does one optimally choose among multiple risky assets? Due to diversi cation, which depends on assets return covariances, the attractiveness
More informationData that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc.
Chapter 8 Measures of Center Data that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc. Data that can only be integer
More information6 Central Limit Theorem. (Chs 6.4, 6.5)
6 Central Limit Theorem (Chs 6.4, 6.5) Motivating Example In the next few weeks, we will be focusing on making statistical inference about the true mean of a population by using sample datasets. Examples?
More informationIMPA Commodities Course : Forward Price Models
IMPA Commodities Course : Forward Price Models Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Department of Statistics and Mathematical Finance Program, University of Toronto, Toronto, Canada http://www.utstat.utoronto.ca/sjaimung
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (34 pts) Answer briefly the following questions. Each question has
More informationSection 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationChapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data
More informationAn Adaptive Time-Step for Increased Model Efficiency
An Adaptive Time-Step for Increased Model Efficiency Todd A. Hutchinson Weather Services International, Andover, Massachusetts 1. Introduction Grid-point numerical weather prediction (NWP) models typically
More informationAnnual risk measures and related statistics
Annual risk measures and related statistics Arno E. Weber, CIPM Applied paper No. 2017-01 August 2017 Annual risk measures and related statistics Arno E. Weber, CIPM 1,2 Applied paper No. 2017-01 August
More informationValue of information of repair times for offshore wind farm maintenance planning
Journal of Physics: Conference Series PAPER OPEN ACCESS Value of information of repair times for offshore wind farm maintenance planning To cite this article: Helene Seyr and Michael Muskulus 2016 J. Phys.:
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2010, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2010, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationMath 14, Homework 7.1 p. 379 # 7, 9, 18, 20, 21, 23, 25, 26 Name
7.1 p. 379 # 7, 9, 18, 0, 1, 3, 5, 6 Name 7. Find each. (a) z α Step 1 Step Shade the desired percent under the mean statistics calculator to 99% confidence interval 3 1 0 1 3 µ 3σ µ σ µ σ µ µ+σ µ+σ µ+3σ
More informationSimple Random Sample
Simple Random Sample A simple random sample (SRS) of size n consists of n elements from the population chosen in such a way that every set of n elements has an equal chance to be the sample actually selected.
More informationChapter 5 Discrete Probability Distributions. Random Variables Discrete Probability Distributions Expected Value and Variance
Chapter 5 Discrete Probability Distributions Random Variables Discrete Probability Distributions Expected Value and Variance.40.30.20.10 0 1 2 3 4 Random Variables A random variable is a numerical description
More informationInventory Models for Special Cases: Multiple Items & Locations
CTL.SC1x -Supply Chain & Logistics Fundamentals Inventory Models for Special Cases: Multiple Items & Locations MIT Center for Transportation & Logistics Agenda Inventory Policies for Multiple Items Grouping
More informationName: CS3130: Probability and Statistics for Engineers Practice Final Exam Instructions: You may use any notes that you like, but no calculators or computers are allowed. Be sure to show all of your work.
More informationEconomics 483. Midterm Exam. 1. Consider the following monthly data for Microsoft stock over the period December 1995 through December 1996:
University of Washington Summer Department of Economics Eric Zivot Economics 3 Midterm Exam This is a closed book and closed note exam. However, you are allowed one page of handwritten notes. Answer all
More informationStatistics for Business and Economics: Random Variables:Continuous
Statistics for Business and Economics: Random Variables:Continuous STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Murray Bourne (interactive
More informationRand Final Pop 2. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: Rand Final Pop 2 Multiple Choice Identify the choice that best completes the statement or answers the question. Scenario 12-1 A high school guidance counselor wonders if it is possible
More informationFinancial Economics. Runs Test
Test A simple statistical test of the random-walk theory is a runs test. For daily data, a run is defined as a sequence of days in which the stock price changes in the same direction. For example, consider
More informationConfidence Intervals. σ unknown, small samples The t-statistic /22
Confidence Intervals σ unknown, small samples The t-statistic 1 /22 Homework Read Sec 7-3. Discussion Question pg 365 Do Ex 7-3 1-4, 6, 9, 12, 14, 15, 17 2/22 Objective find the confidence interval for
More information1 Inferential Statistic
1 Inferential Statistic Population versus Sample, parameter versus statistic A population is the set of all individuals the researcher intends to learn about. A sample is a subset of the population and
More informationPRMIA Exam 8002 PRM Certification - Exam II: Mathematical Foundations of Risk Measurement Version: 6.0 [ Total Questions: 132 ]
s@lm@n PRMIA Exam 8002 PRM Certification - Exam II: Mathematical Foundations of Risk Measurement Version: 6.0 [ Total Questions: 132 ] Question No : 1 A 2-step binomial tree is used to value an American
More informationIn April 2013, the UK government brought into force a tax on carbon
The UK carbon floor and power plant hedging Due to the carbon floor, the price of carbon emissions has become a highly significant part of the generation costs for UK power producers. Vytautas Jurenas
More informationDiscounting a mean reverting cash flow
Discounting a mean reverting cash flow Marius Holtan Onward Inc. 6/26/2002 1 Introduction Cash flows such as those derived from the ongoing sales of particular products are often fluctuating in a random
More informationCh. 8 Risk and Rates of Return. Return, Risk and Capital Market. Investment returns
Ch. 8 Risk and Rates of Return Topics Measuring Return Measuring Risk Risk & Diversification CAPM Return, Risk and Capital Market Managers must estimate current and future opportunity rates of return for
More informationMAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
More informationINSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS. 20 th May Subject CT3 Probability & Mathematical Statistics
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 20 th May 2013 Subject CT3 Probability & Mathematical Statistics Time allowed: Three Hours (10.00 13.00) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1.
More informationAppendix S: Content Portfolios and Diversification
Appendix S: Content Portfolios and Diversification 1188 The expected return on a portfolio is a weighted average of the expected return on the individual id assets; but estimating the risk, or standard
More informationDiploma Part 2. Quantitative Methods. Examiner s Suggested Answers
Diploma Part 2 Quantitative Methods Examiner s Suggested Answers Question 1 (a) The binomial distribution may be used in an experiment in which there are only two defined outcomes in any particular trial
More informationTechnical risks on various stages of a project
Technical risks on various stages of a project Pure experience in onshore wind energy 19 May 2017 1 SAFER, SMARTER, GREENER MSc. Maciej Puto 2 OPERATION DECOMMISSIONING EXECUTION DEVELOPMENT PRE-DEVELOPMENT
More informationInput to new IEC design standards: Two case studies of iced turbine load analysis
Input to new IEC 61400-1 design standards: Two case studies of iced turbine load analysis Presented by: Ville Lehtomäki, VTT Technical Research Centre of Finland February 10-11 2014; Sundsvall, Sweden
More information1 Assises de l Energie 2007 Liège, 10-11/10/2007 Offshore wind energy in the North Sea Overview
Offshore wind energy in the North Sea Ronnie Belmans Energiecongres 2007 Overview Why offshore? International status Situation in Belgium Elia network Supergrid Conclusion www.ef4.be 1 Why offshore? Wind
More informationSTAT 111 Recitation 4
STAT 111 Recitation 4 Linjun Zhang http://stat.wharton.upenn.edu/~linjunz/ September 29, 2017 Misc. Mid-term exam time: 6-8 pm, Wednesday, Oct. 11 The mid-term break is Oct. 5-8 The next recitation class
More informationCorrecting for Survival Effects in Cross Section Wage Equations Using NBA Data
Correcting for Survival Effects in Cross Section Wage Equations Using NBA Data by Peter A Groothuis Professor Appalachian State University Boone, NC and James Richard Hill Professor Central Michigan University
More information7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4 - Would the correlation between x and y in the table above be positive or negative? The correlation is negative. -
More informationDistribution of the Sample Mean
Distribution of the Sample Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Experiment (1 of 3) Suppose we have the following population : 4 8 1 2 3 4 9 1
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationMonte Carlo Introduction
Monte Carlo Introduction Probability Based Modeling Concepts moneytree.com Toll free 1.877.421.9815 1 What is Monte Carlo? Monte Carlo Simulation is the currently accepted term for a technique used by
More informationWe believe our human capital is the key factor to success, as well as the reason r upholding a leading position on the Bulgarian financial market.
ELANA is a Bulgarian financial non-banking institution with over 16 years history in setting the novelties on the local financial market in transition n relying on being stable traditional partner for
More informationσ e, which will be large when prediction errors are Linear regression model
Linear regression model we assume that two quantitative variables, x and y, are linearly related; that is, the population of (x, y) pairs are related by an ideal population regression line y = α + βx +
More informationFINC 430 TA Session 7 Risk and Return Solutions. Marco Sammon
FINC 430 TA Session 7 Risk and Return Solutions Marco Sammon Formulas for return and risk The expected return of a portfolio of two risky assets, i and j, is Expected return of asset - the percentage of
More informationMAS187/AEF258. University of Newcastle upon Tyne
MAS187/AEF258 University of Newcastle upon Tyne 2005-6 Contents 1 Collecting and Presenting Data 5 1.1 Introduction...................................... 5 1.1.1 Examples...................................
More informationGame Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati.
Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati. Module No. # 06 Illustrations of Extensive Games and Nash Equilibrium
More informationOn modelling of electricity spot price
, Rüdiger Kiesel and Fred Espen Benth Institute of Energy Trading and Financial Services University of Duisburg-Essen Centre of Mathematics for Applications, University of Oslo 25. August 2010 Introduction
More informationCriteria for rating wind power projects
Criteria for rating wind power projects Executive Summary CRISIL has outstanding ratings on 21 wind power project companies as on June 30, 2015. Wind power projects depend primarily on wind speeds for
More informationBuilding Your Proforma
Building Your Proforma A Million Little Pieces Community Wind Energy 2006 Expanding The Know-how - Expanding The Market Iowa Events Center, Des Moines, Iowa March 7-8, 2006 Thomas A. Wind, PE Wind Utility
More informationConover Test of Variances (Simulation)
Chapter 561 Conover Test of Variances (Simulation) Introduction This procedure analyzes the power and significance level of the Conover homogeneity test. This test is used to test whether two or more population
More informationDavid Tenenbaum GEOG 090 UNC-CH Spring 2005
Simple Descriptive Statistics Review and Examples You will likely make use of all three measures of central tendency (mode, median, and mean), as well as some key measures of dispersion (standard deviation,
More informationIndependent Power Producer with 33.2 MWs of Awarded ComFIT Projects in NS
Irish Mountain 6.0 MW 2014 project Watts Section 1.5 MW In-service: 2011 Wedgeport 1.6 MW 2014 project Shag Harbour 3.2 MW 2014 project Watts Wind Current and Phase One Projects Independent Power Producer
More informationAn Introduction to Statistical Extreme Value Theory
An Introduction to Statistical Extreme Value Theory Uli Schneider Geophysical Statistics Project, NCAR January 26, 2004 NCAR Outline Part I - Two basic approaches to extreme value theory block maxima,
More informationLecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 7.4-1
Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Section 7.4-1 Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7- Estimating a Population
More informationA continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
More informationAn Independent Power Producer with 33 MWs of Awarded ComFIT Projects in Nova Scotia at $0.131/kW
An Independent Power Producer with 33 MWs of Awarded ComFIT Projects in Nova Scotia at $0.131/kW 1 DISCLAIMER Watts Wind Energy Inc. Important Notice: This document has been prepared by Watts Wind Energy
More informationPHASE I.A. STOCHASTIC STUDY TESTIMONY OF DR. SHUCHENG LIU ON BEHALF OF THE CALIFORNIA INDEPENDENT SYSTEM OPERATOR CORPORATION
Rulemaking No.: 13-12-010 Exhibit No.: Witness: Dr. Shucheng Liu Order Instituting Rulemaking to Integrate and Refine Procurement Policies and Consider Long-Term Procurement Plans. Rulemaking 13-12-010
More informationORE Applied: Dynamic Initial Margin and MVA
ORE Applied: Dynamic Initial Margin and MVA Roland Lichters QuantLib User Meeting at IKB, Düsseldorf 8 December 2016 Agenda Open Source Risk Engine Dynamic Initial Margin and Margin Value Adjustment Conclusion
More informationCMA Workforce Survey Methodology. Objective
CMA Workforce Survey 2017 Methodology Objective The CMA Workforce Survey aimed to collect information from physicians on a wide range of topics relating to their practice in Canada; including but not limited
More information