Laws of Total Expectation and Total Variance
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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. -
Probability Cheatsheet V2.0 Thinking Conditionally Law of Total Probability (LOTP)
Probability Cheatsheet v2.0 Thinking Conditionally Law of Total Probability (LOTP) Let B1;B2;B3; :::Bn be a partition of the sample space (i.e., they are Compiled by William Chen (http://wzchen.com) and Joe Blitzstein, Independence disjoint and their union is the entire sample space). with contributions from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Independent Events A and B are independent if knowing whether P (A) = P (AjB )P (B ) + P (AjB )P (B ) + ··· + P (AjB )P (B ) Jessy Hwang. Material based on Joe Blitzstein's (@stat110) lectures 1 1 2 2 n n A occurred gives no information about whether B occurred. More (http://stat110.net) and Blitzstein/Hwang's Introduction to P (A) = P (A \ B1) + P (A \ B2) + ··· + P (A \ Bn) formally, A and B (which have nonzero probability) are independent if Probability textbook (http://bit.ly/introprobability). Licensed and only if one of the following equivalent statements holds: For LOTP with extra conditioning, just add in another event C! under CC BY-NC-SA 4.0. Please share comments, suggestions, and errors at http://github.com/wzchen/probability_cheatsheet. P (A \ B) = P (A)P (B) P (AjC) = P (AjB1;C)P (B1jC) + ··· + P (AjBn;C)P (BnjC) P (AjB) = P (A) P (AjC) = P (A \ B1jC) + P (A \ B2jC) + ··· + P (A \ BnjC) P (BjA) = P (B) Last Updated September 4, 2015 Special case of LOTP with B and Bc as partition: Conditional Independence A and B are conditionally independent P (A) = P (AjB)P (B) + P (AjBc)P (Bc) given C if P (A \ BjC) = P (AjC)P (BjC). -
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. -
Probability Cheatsheet
Probability Cheatsheet v1.1.1 Simpson's Paradox Expected Value, Linearity, and Symmetry P (A j B; C) < P (A j Bc;C) and P (A j B; Cc) < P (A j Bc;Cc) Expected Value (aka mean, expectation, or average) can be thought Compiled by William Chen (http://wzchen.com) with contributions yet still, P (A j B) > P (A j Bc) of as the \weighted average" of the possible outcomes of our random from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Jessy Hwang. variable. Mathematically, if x1; x2; x3;::: are all of the possible values Material based off of Joe Blitzstein's (@stat110) lectures Bayes' Rule and Law of Total Probability that X can take, the expected value of X can be calculated as follows: P (http://stat110.net) and Blitzstein/Hwang's Intro to Probability E(X) = xiP (X = xi) textbook (http://bit.ly/introprobability). Licensed under CC i Law of Total Probability with partitioning set B1; B2; B3; :::Bn and BY-NC-SA 4.0. Please share comments, suggestions, and errors at with extra conditioning (just add C!) Note that for any X and Y , a and b scaling coefficients and c is our http://github.com/wzchen/probability_cheatsheet. constant, the following property of Linearity of Expectation holds: P (A) = P (AjB1)P (B1) + P (AjB2)P (B2) + :::P (AjBn)P (Bn) Last Updated March 20, 2015 P (A) = P (A \ B1) + P (A \ B2) + :::P (A \ Bn) E(aX + bY + c) = aE(X) + bE(Y ) + c P (AjC) = P (AjB1; C)P (B1jC) + :::P (AjBn; C)P (BnjC) If two Random Variables have the same distribution, even when they are dependent by the property of Symmetry their expected values P (AjC) = P (A \ B jC) + P (A \ B jC) + :::P (A \ B jC) Counting 1 2 n are equal. -
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. -
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). -
Chapter 3: Expectation and Variance
44 Chapter 3: Expectation and Variance In the previous chapter we looked at probability, with three major themes: 1. Conditional probability: P(A | B). 2. First-step analysis for calculating eventual probabilities in a stochastic process. 3. Calculating probabilities for continuous and discrete random variables. In this chapter, we look at the same themes for expectation and variance. The expectation of a random variable is the long-term average of the ran- dom variable. Imagine observing many thousands of independent random values from the random variable of interest. Take the average of these random values. The expectation is the value of this average as the sample size tends to infinity. We will repeat the three themes of the previous chapter, but in a different order. 1. Calculating expectations for continuous and discrete random variables. 2. Conditional expectation: the expectation of a random variable X, condi- tional on the value taken by another random variable Y . If the value of Y affects the value of X (i.e. X and Y are dependent), the conditional expectation of X given the value of Y will be different from the overall expectation of X. 3. First-step analysis for calculating the expected amount of time needed to reach a particular state in a process (e.g. the expected number of shots before we win a game of tennis). We will also study similar themes for variance. 45 3.1 Expectation The mean, expected value, or expectation of a random variable X is writ- ten as E(X) or µX . If we observe N random values of X, then the mean of the N values will be approximately equal to E(X) for large N. -
Lecture Notes 4 Expectation • Definition and Properties
Lecture Notes 4 Expectation • Definition and Properties • Covariance and Correlation • Linear MSE Estimation • Sum of RVs • Conditional Expectation • Iterated Expectation • Nonlinear MSE Estimation • Sum of Random Number of RVs Corresponding pages from B&T: 81-92, 94-98, 104-115, 160-163, 171-174, 179, 225-233, 236-247. EE 178/278A: Expectation Page 4–1 Definition • We already introduced the notion of expectation (mean) of a r.v. • We generalize this definition and discuss it in more depth • Let X ∈X be a discrete r.v. with pmf pX(x) and g(x) be a function of x. The expectation or expected value of g(X) is defined as E(g(X)) = g(x)pX(x) x∈X • For a continuous r.v. X ∼ fX(x), the expected value of g(X) is defined as ∞ E(g(X)) = g(x)fX(x) dx −∞ • Examples: ◦ g(X)= c, a constant, then E(g(X)) = c ◦ g(X)= X, E(X)= x xpX(x) is the mean of X k k ◦ g(X)= X , E(X ) is the kth moment of X ◦ g(X)=(X − E(X))2, E (X − E(X))2 is the variance of X EE 178/278A: Expectation Page 4–2 • Expectation is linear, i.e., for any constants a and b E[ag1(X)+ bg2(X)] = a E(g1(X)) + b E(g2(X)) Examples: ◦ E(aX + b)= a E(X)+ b ◦ Var(aX + b)= a2Var(X) Proof: From the definition Var(aX + b) = E ((aX + b) − E(aX + b))2 = E (aX + b − a E(X) − b)2 = E a2(X − E(X))2 = a2E (X − E(X))2 = a2Var( X) EE 178/278A: Expectation Page 4–3 Fundamental Theorem of Expectation • Theorem: Let X ∼ pX(x) and Y = g(X) ∼ pY (y), then E(Y )= ypY (y)= g(x)pX(x)=E(g(X)) y∈Y x∈X • The same formula holds for fY (y) using integrals instead of sums • Conclusion: E(Y ) can be found using either fX(x) or fY (y). -
Assumptions of the Simple Classical Linear Regression Model (CLRM)
ECONOMICS 351* -- NOTE 1 M.G. Abbott ECON 351* -- NOTE 1 Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction CLRM stands for the Classical Linear Regression Model. The CLRM is also known as the standard linear regression model. Three sets of assumptions define the CLRM. 1. Assumptions respecting the formulation of the population regression equation, or PRE. Assumption A1 2. Assumptions respecting the statistical properties of the random error term and the dependent variable. Assumptions A2-A4 3. Assumptions respecting the properties of the sample data. Assumptions A5-A8 ECON 351* -- Note 1: Specification of the Simple CLRM … Page 1 of 16 pages ECONOMICS 351* -- NOTE 1 M.G. Abbott • Figure 2.1 Plot of Population Data Points, Conditional Means E(Y|X), and the Population Regression Function PRF Y Fitted values 200 $ PRF = 175 e, ur β0 + β1Xi t i nd 150 pe ex n o i t 125 p um 100 cons y ekl e W 75 50 60 80 100 120 140 160 180 200 220 240 260 Weekly income, $ Recall that the solid line in Figure 2.1 is the population regression function, which takes the form f (Xi ) = E(Yi Xi ) = β0 + β1Xi . For each population value Xi of X, there is a conditional distribution of population Y values and a corresponding conditional distribution of population random errors u, where (1) each population value of u for X = Xi is u i X i = Yi − E(Yi X i ) = Yi − β0 − β1 X i , and (2) each population value of Y for X = Xi is Yi X i = E(Yi X i ) + u i = β0 + β1 X i + u i . -
Probabilities, Random Variables and Distributions A
Probabilities, Random Variables and Distributions A Contents A.1 EventsandProbabilities................................ 318 A.1.1 Conditional Probabilities and Independence . ............. 318 A.1.2 Bayes’Theorem............................... 319 A.2 Random Variables . ................................. 319 A.2.1 Discrete Random Variables ......................... 319 A.2.2 Continuous Random Variables ....................... 320 A.2.3 TheChangeofVariablesFormula...................... 321 A.2.4 MultivariateNormalDistributions..................... 323 A.3 Expectation,VarianceandCovariance........................ 324 A.3.1 Expectation................................. 324 A.3.2 Variance................................... 325 A.3.3 Moments................................... 325 A.3.4 Conditional Expectation and Variance ................... 325 A.3.5 Covariance.................................. 326 A.3.6 Correlation.................................. 327 A.3.7 Jensen’sInequality............................. 328 A.3.8 Kullback–LeiblerDiscrepancyandInformationInequality......... 329 A.4 Convergence of Random Variables . 329 A.4.1 Modes of Convergence . 329 A.4.2 Continuous Mapping and Slutsky’s Theorem . 330 A.4.3 LawofLargeNumbers........................... 330 A.4.4 CentralLimitTheorem........................... 331 A.4.5 DeltaMethod................................ 331 A.5 ProbabilityDistributions............................... 332 A.5.1 UnivariateDiscreteDistributions...................... 333 A.5.2 Univariate Continuous Distributions . 335 -
Lectures Prepared By: Elchanan Mossel Yelena Shvets
Introduction to probability Stat 134 FAll 2005 Berkeley Lectures prepared by: Elchanan Mossel Yelena Shvets Follows Jim Pitman’s book: Probability Sections 6.1-6.2 # of Heads in a Random # of Tosses •Suppose a fair die is rolled and let N be the number on top. N=5 •Next a fair coin is tossed N times and H, the number of heads is recorded: H=3 Question : Find the distribution of H, the number of heads? # of Heads in a Random Number of Tosses Solution : •The conditional probability for the H=h , given N=n is •By the rule of the average conditional probability and using P(N=n) = 1/6 h 0 1 2 3 4 5 6 P(H=h) 63/384 120/384 99/384 64/384 29/384 8/384 1/384 Conditional Distribution of Y given X =x Def: For each value x of X, the set of probabilities P(Y=y|X=x) where y varies over all possible values of Y, form a probability distribution, depending on x, called the conditional distribution of Y given X=x . Remarks: •In the example P(H = h | N =n) ~ binomial(n,½). •The unconditional distribution of Y is the average of the conditional distributions, weighted by P(X=x). Conditional Distribution of X given Y=y Remark: Once we have the conditional distribution of Y given X=x , using Bayes’ rule we may obtain the conditional distribution of X given Y=y . •Example : We have computed distribution of H given N = n : •Using the product rule we can get the joint distr.