Introduction to Probability: Lecture 13: Conditional Expectation & Variance
Total Page：16
File Type：pdf, Size：1020Kb
Load more
Recommended publications
-
Lecture 22: Bivariate Normal Distribution Distribution
6.5 Conditional Distributions General Bivariate Normal Let Z1; Z2 ∼ N (0; 1), which we will use to build a general bivariate normal Lecture 22: Bivariate Normal Distribution distribution. 1 1 2 2 f (z1; z2) = exp − (z1 + z2 ) Statistics 104 2π 2 We want to transform these unit normal distributions to have the follow Colin Rundel arbitrary parameters: µX ; µY ; σX ; σY ; ρ April 11, 2012 X = σX Z1 + µX p 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 1 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - Marginals General Bivariate Normal - Cov/Corr First, lets examine the marginal distributions of X and Y , Second, we can find Cov(X ; Y ) and ρ(X ; Y ) Cov(X ; Y ) = E [(X − E(X ))(Y − E(Y ))] X = σX Z1 + µX h p i = E (σ Z + µ − µ )(σ [ρZ + 1 − ρ2Z ] + µ − µ ) = σX N (0; 1) + µX X 1 X X Y 1 2 Y Y 2 h p 2 i = N (µX ; σX ) = E (σX Z1)(σY [ρZ1 + 1 − ρ Z2]) h 2 p 2 i = σX σY E ρZ1 + 1 − ρ Z1Z2 p 2 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY = σX σY ρE[Z1 ] p 2 = σX σY ρ = σY [ρN (0; 1) + 1 − ρ N (0; 1)] + µY = σ [N (0; ρ2) + N (0; 1 − ρ2)] + µ Y Y Cov(X ; Y ) ρ(X ; Y ) = = ρ = σY N (0; 1) + µY σX σY 2 = N (µY ; σY ) Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 2 / 22 Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 3 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - RNG Multivariate Change of Variables Consequently, if we want to generate a Bivariate Normal random variable Let X1;:::; Xn have a continuous joint distribution with pdf f defined of S. -
Ffontiau Cymraeg
This publication is available in other languages and formats on request. Mae'r cyhoeddiad hwn ar gael mewn ieithoedd a fformatau eraill ar gais. [email protected] www.caerphilly.gov.uk/equalities How to type Accented Characters This guidance document has been produced to provide practical help when typing letters or circulars, or when designing posters or flyers so that getting accents on various letters when typing is made easier. The guide should be used alongside the Council’s Guidance on Equalities in Designing and Printing. Please note this is for PCs only and will not work on Macs. Firstly, on your keyboard make sure the Num Lock is switched on, or the codes shown in this document won’t work (this button is found above the numeric keypad on the right of your keyboard). By pressing the ALT key (to the left of the space bar), holding it down and then entering a certain sequence of numbers on the numeric keypad, it's very easy to get almost any accented character you want. For example, to get the letter “ô”, press and hold the ALT key, type in the code 0 2 4 4, then release the ALT key. The number sequences shown from page 3 onwards work in most fonts in order to get an accent over “a, e, i, o, u”, the vowels in the English alphabet. In other languages, for example in French, the letter "c" can be accented and in Spanish, "n" can be accented too. Many other languages have accents on consonants as well as vowels. -
Laws of Total Expectation and Total Variance
Laws of Total Expectation and Total Variance Definition of conditional density. Assume and arbitrary random variable X with density fX . Take an event A with P (A) > 0. Then the conditional density fXjA is defined as follows: 8 < f(x) 2 P (A) x A fXjA(x) = : 0 x2 = A Note that the support of fXjA is supported only in A. Definition of conditional expectation conditioned on an event. Z Z 1 E(h(X)jA) = h(x) fXjA(x) dx = h(x) fX (x) dx A P (A) A Example. For the random variable X with density function 8 < 1 3 64 x 0 < x < 4 f(x) = : 0 otherwise calculate E(X2 j X ≥ 1). Solution. Step 1. Z h i 4 1 1 1 x=4 255 P (X ≥ 1) = x3 dx = x4 = 1 64 256 4 x=1 256 Step 2. Z Z ( ) 1 256 4 1 E(X2 j X ≥ 1) = x2 f(x) dx = x2 x3 dx P (X ≥ 1) fx≥1g 255 1 64 Z ( ) ( )( ) h i 256 4 1 256 1 1 x=4 8192 = x5 dx = x6 = 255 1 64 255 64 6 x=1 765 Definition of conditional variance conditioned on an event. Var(XjA) = E(X2jA) − E(XjA)2 1 Example. For the previous example , calculate the conditional variance Var(XjX ≥ 1) Solution. We already calculated E(X2 j X ≥ 1). We only need to calculate E(X j X ≥ 1). Z Z Z ( ) 1 256 4 1 E(X j X ≥ 1) = x f(x) dx = x x3 dx P (X ≥ 1) fx≥1g 255 1 64 Z ( ) ( )( ) h i 256 4 1 256 1 1 x=4 4096 = x4 dx = x5 = 255 1 64 255 64 5 x=1 1275 Finally: ( ) 8192 4096 2 630784 Var(XjX ≥ 1) = E(X2jX ≥ 1) − E(XjX ≥ 1)2 = − = 765 1275 1625625 Definition of conditional expectation conditioned on a random variable. -
Form IT-2658-E:12/19:Certificate of Exemption from Partnership Or New
Department of Taxation and Finance IT-2658-E Certificate of Exemption from Partnership (12/19) or New York S Corporation Estimated Tax Paid on Behalf of Nonresident Individual Partners and Shareholders Do not send this certificate to the Tax Department (see instructions below). Use this certificate for tax years 2020 and 2021; it will expire on February 1, 2022. First name and middle initial Last name Social Security number Mailing address (number and street or PO box) Telephone number ( ) City, village, or post office State ZIP code I certify that I will comply with the New York State estimated tax provisions and tax return filing requirements, to the extent that they apply to me, for tax years 2020 and 2021 (see instructions). Signature of nonresident individual partner or shareholder Date Instructions General information If you do not meet either of the above exceptions, you may still claim exemption from this estimated tax provision by Tax Law section 658(c)(4) requires the following entities filing Form IT-2658-E. that have income derived from New York sources to make estimated personal income tax payments on behalf of You qualify to claim exemption and file Form IT-2658-E by partners or shareholders who are nonresident individuals: certifying that you will comply in your individual capacity with • New York S corporations; all the New York State personal income tax and MCTMT estimated tax and tax return filing requirements, to the • partnerships (other than publicly traded partnerships as extent that they apply to you, for the years covered by this defined in Internal Revenue Code section 7704); and certificate. -
Left Ventricular Diastolic Dysfunction in Ischemic Stroke: Functional and Vascular Outcomes
Journal of Stroke 2016;18(2):195-202 http://dx.doi.org/10.5853/jos.2015.01669 Original Article Left Ventricular Diastolic Dysfunction in Ischemic Stroke: Functional and Vascular Outcomes Hong-Kyun Park,a* Beom Joon Kim,a* Chang-Hwan Yoon,b Mi Hwa Yang,a Moon-Ku Han,a Hee-Joon Baea aDepartment of Neurology and Cerebrovascular Center, Seoul National University Bundang Hospital, Seongnam, Korea bDivision of Cardiology, Department of Internal Medicine, Seoul National University Bundang Hospital, Seongnam, Korea Background and Purpose Left ventricular (LV) diastolic dysfunction, developed in relation to myo- Correspondence: Moon-Ku Han cardial dysfunction and remodeling, is documented in 15%-25% of the population. However, its role Department of Neurology and Cerebrovascular Center, Seoul National in functional recovery and recurrent vascular events after acute ischemic stroke has not been thor- University Bundang Hospital, oughly investigated. 82 Gumi-ro 173beon-gil, Bundang-gu, Seongnam 13620, Korea Methods In this retrospective observational study, we identified 2,827 ischemic stroke cases with Tel: +82-31-787-7464 adequate echocardiographic evaluations to assess LV diastolic dysfunction within 1 month after the Fax: +82-31-787-4059 index stroke. The peak transmitral filling velocity/mean mitral annular velocity during early diastole (E/ E-mail: [email protected] e’) was used to estimate LV diastolic dysfunction. We divided patients into 3 groups according to E/e’ Received: December 1, 2015 as follows: <8, 8-15, and ≥15. Recurrent vascular events and functional recovery were prospectively Revised: March 14, 2016 collected at 3 months and 1 year. Accepted: March 21, 2016 Results Among included patients, E/e’ was 10.6±6.4: E/e’ <8 in 993 (35%), 8-15 in 1,444 (51%), and *These authors contributed equally to this ≥15 in 378 (13%) cases. -
Multicollinearity
Chapter 10 Multicollinearity 10.1 The Nature of Multicollinearity 10.1.1 Extreme Collinearity The standard OLS assumption that ( xi1; xi2; : : : ; xik ) not be linearly related means that for any ( c1; c2; : : : ; ck ) xik =6 c1xi1 + c2xi2 + ··· + ck 1xi;k 1 (10.1) − − for some i. If the assumption is violated, then we can find ( c1; c2; : : : ; ck 1 ) such that − xik = c1xi1 + c2xi2 + ··· + ck 1xi;k 1 (10.2) − − for all i. Define x12 ··· x1k xk1 c1 x22 ··· x2k xk2 c2 X1 = 0 . 1 ; xk = 0 . 1 ; and c = 0 . 1 : . B C B C B C B xn2 ··· xnk C B xkn C B ck 1 C B C B C B − C @ A @ A @ A Then extreme collinearity can be represented as xk = X1c: (10.3) We have represented extreme collinearity in terms of the last explanatory vari- able. Since we can always re-order the variables this choice is without loss of generality and the analysis could be applied to any non-constant variable by moving it to the last column. 10.1.2 Near Extreme Collinearity Of course, it is rare, in practice, that an exact linear relationship holds. Instead, we have xik = c1xi1 + c2xi2 + ··· + ck 1xi;k 1 + vi (10.4) − − 133 134 CHAPTER 10. MULTICOLLINEARITY or, more compactly, xk = X1c + v; (10.5) where the v's are small relative to the x's. If we think of the v's as random variables they will have small variance (and zero mean if X includes a column of ones). A convenient way to algebraically express the degree of collinearity is the sample correlation between xik and wi = c1xi1 +c2xi2 +···+ck 1xi;k 1, namely − − cov( xik; wi ) cov( wi + vi; wi ) rx;w = = (10.6) var(xi;k)var(wi) var(wi + vi)var(wi) d d Clearly, as the variancep of vi grows small, thisp value will go to unity. -
Typing in Greek Sarah Abowitz Smith College Classics Department
Typing in Greek Sarah Abowitz Smith College Classics Department Windows 1. Down at the lower right corner of the screen, click the letters ENG, then select Language Preferences in the pop-up menu. If these letters are not present at the lower right corner of the screen, open Settings, click on Time & Language, then select Region & Language in the sidebar to get to the proper screen for step 2. 2. When this window opens, check if Ελληνικά/Greek is in the list of keyboards on your computer under Languages. If so, go to step 3. Otherwise, click Add A New Language. Clicking Add A New Language will take you to this window. Look for Ελληνικά/Greek and click it. When you click Ελληνικά/Greek, the language will be added and you will return to the previous screen. 3. Now that Ελληνικά is listed in your computer’s languages, click it and then click Options. 4. Click Add A Keyboard and add the Greek Polytonic option. If you started this tutorial without the pictured keyboard menu in step 1, it should be in the lower right corner of your screen now. 5. To start typing in Greek, click the letters ENG next to the clock in the lower right corner of the screen. Choose “Greek Polytonic keyboard” to start typing in greek, and click “US keyboard” again to go back to English. Mac 1. Click the apple button in the top left corner of your screen. From the drop-down menu, choose System Preferences. When the window below appears, click the “Keyboard” icon. -
CONDITIONAL EXPECTATION Definition 1. Let (Ω,F,P)
CONDITIONAL EXPECTATION 1. CONDITIONAL EXPECTATION: L2 THEORY ¡ Definition 1. Let (,F ,P) be a probability space and let G be a σ algebra contained in F . For ¡ any real random variable X L2(,F ,P), define E(X G ) to be the orthogonal projection of X 2 j onto the closed subspace L2(,G ,P). This definition may seem a bit strange at first, as it seems not to have any connection with the naive definition of conditional probability that you may have learned in elementary prob- ability. However, there is a compelling rationale for Definition 1: the orthogonal projection E(X G ) minimizes the expected squared difference E(X Y )2 among all random variables Y j ¡ 2 L2(,G ,P), so in a sense it is the best predictor of X based on the information in G . It may be helpful to consider the special case where the σ algebra G is generated by a single random ¡ variable Y , i.e., G σ(Y ). In this case, every G measurable random variable is a Borel function Æ ¡ of Y (exercise!), so E(X G ) is the unique Borel function h(Y ) (up to sets of probability zero) that j minimizes E(X h(Y ))2. The following exercise indicates that the special case where G σ(Y ) ¡ Æ for some real-valued random variable Y is in fact very general. Exercise 1. Show that if G is countably generated (that is, there is some countable collection of set B G such that G is the smallest σ algebra containing all of the sets B ) then there is a j 2 ¡ j G measurable real random variable Y such that G σ(Y ). -
Conditional Expectation and Prediction Conditional Frequency Functions and Pdfs Have Properties of Ordinary Frequency and Density Functions
Conditional expectation and prediction Conditional frequency functions and pdfs have properties of ordinary frequency and density functions. Hence, associated with a conditional distribution is a conditional mean. Y and X are discrete random variables, the conditional frequency function of Y given x is pY|X(y|x). Conditional expectation of Y given X=x is Continuous case: Conditional expectation of a function: Consider a Poisson process on [0, 1] with mean λ, and let N be the # of points in [0, 1]. For p < 1, let X be the number of points in [0, p]. Find the conditional distribution and conditional mean of X given N = n. Make a guess! Consider a Poisson process on [0, 1] with mean λ, and let N be the # of points in [0, 1]. For p < 1, let X be the number of points in [0, p]. Find the conditional distribution and conditional mean of X given N = n. We first find the joint distribution: P(X = x, N = n), which is the probability of x events in [0, p] and n−x events in [p, 1]. From the assumption of a Poisson process, the counts in the two intervals are independent Poisson random variables with parameters pλ and (1−p)λ (why?), so N has Poisson marginal distribution, so the conditional frequency function of X is Binomial distribution, Conditional expectation is np. Conditional expectation of Y given X=x is a function of X, and hence also a random variable, E(Y|X). In the last example, E(X|N=n)=np, and E(X|N)=Np is a function of N, a random variable that generally has an expectation Taken w.r.t. -
Kevin E. O'malley
KEVIN E. O’MALLEY Shareholder Gallagher & Kennedy 2575 E. Camelback Road, Suite 1100 Phoenix, AZ 85016 DIR: 602-530-8430 [email protected] My Practice INSURANCE - Coverage Analysis and Litigation for Insurance Companies, Insurance Disputes, Insurance for Individuals and Businesses PUBLIC BIDDING & PROCUREMENT - Business Law and Transactions, Claims, Disputes and Litigation, Compliance, Construction, Contract Negotiations, Government Affairs and Lobbying, Government Bid and Proposal Preparation and Submission, Intellectual Property ADMINISTRATIVE LAW - Public Bidding and Procurement Administrative Compliance PROFESSIONAL LIABILITY LITIGATION - Civil Defense Litigation, Class Actions, PB&P Claims, Disputes and Litigation, Real Estate Litigation CONSTRUCTION - Construction Litigation, Construction Public Bidding and Procurement PROFESSIONAL SUMMARY Kevin heads Gallagher & Kennedy’s litigation and public bidding and procurement departments. Keenly attuned to the needs of his wide array of clients, Kevin offers proven skill in developing strategy and implementing solutions that are singularly focused on each client’s often unique and/or specific goals. Over his 30-year career representing public and private companies across a wide range of industries, Kevin’s litigation skills have been honed to a razor’s edge. Notably, Kevin’s civil litigation experience is extensive and far reaching, both in and out of the courtroom. Kevin has managed many high profile and complex matters, including securities, commercial, environmental, governmental, construction, professional liability and insurance litigation cases. He offers significant class action and appellate experience. A veteran litigator and talented negotiator with a seemingly natural aptitude for advocacy and unwavering drive to win, Kevin is appreciated for his strategic, tactical and streamlined arguments and for his ability to transform client positions into compelling and easily understood talking points. -
I/B/E/S DETAIL HISTORY a Guide to the Analyst-By-Analyst Historical Earnings Estimate Database
I/B/E/S DETAIL HISTORY A guide to the analyst-by-analyst historical earnings estimate database U.S. Edition CONTENTS page Overview..................................................................................................................................................................................................................1 File Explanations ..............................................................................................................................................................................................2 Detail File ..................................................................................................................................................................................................2 Identifier File..........................................................................................................................................................................................3 Adjustments File..................................................................................................................................................................................3 Excluded Estimates File ...............................................................................................................................................................3 Broker Translations...........................................................................................................................................................................4 S/I/G Codes..............................................................................................................................................................................................4 -
Section 1: Regression Review
Section 1: Regression review Yotam Shem-Tov Fall 2015 Yotam Shem-Tov STAT 239A/ PS 236A September 3, 2015 1 / 58 Contact information Yotam Shem-Tov, PhD student in economics. E-mail: [email protected]. Office: Evans hall room 650 Office hours: To be announced - any preferences or constraints? Feel free to e-mail me. Yotam Shem-Tov STAT 239A/ PS 236A September 3, 2015 2 / 58 R resources - Prior knowledge in R is assumed In the link here is an excellent introduction to statistics in R . There is a free online course in coursera on programming in R. Here is a link to the course. An excellent book for implementing econometric models and method in R is Applied Econometrics with R. This is my favourite for starting to learn R for someone who has some background in statistic methods in the social science. The book Regression Models for Data Science in R has a nice online version. It covers the implementation of regression models using R . A great link to R resources and other cool programming and econometric tools is here. Yotam Shem-Tov STAT 239A/ PS 236A September 3, 2015 3 / 58 Outline There are two general approaches to regression 1 Regression as a model: a data generating process (DGP) 2 Regression as an algorithm (i.e., an estimator without assuming an underlying structural model). Overview of this slides: 1 The conditional expectation function - a motivation for using OLS. 2 OLS as the minimum mean squared error linear approximation (projections). 3 Regression as a structural model (i.e., assuming the conditional expectation is linear).