# 20110921 class notes

## Computer and Network Security: MIT 6.857 Class Notes pptx

... 20 60 1000 2500 93000 MIT 6.857 Computer and Network Security Class Notes File: http://theory.lcs .mit. edu/˜rivest /notes/ divisors.pdf Revision: December 2, 2002 Divisors and Divisibility Deﬁnition ... freely reproduced for educational or personal use MIT 6.857 Computer and Network Security Class Notes File: http://theory.lcs .mit. edu/˜rivest /notes/ bignum.pdf Revision: December 2, 2002 Bignum ... MIT 6.857 Computer and Network Security Class Notes File: http://theory.lcs .mit. edu/˜rivest /notes/ ntintro.pdf Revision: December 2, 2002 Introduction...
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## Capital Markets and Portfolio TheoryRoland Portait From the class notes taken by Peng pptx

... The market value of asset i is Vi , then (by deịnition of the market portfolio) its weight in the market portfolio is: Vi mi = PN i=1 Vi The return of the market portfolio is: RM = m0 R Hypothesis ... non- Chapter Portfolio Choices Proof Assume that the assets are redundant, then there exist N scalars , , , N such that PN i=1 i Ri = k Consider the portfolio deịned by the weights The variance ... 10 à ã1 = 1 Then the solution of (P ) writes: x = k1 + b k Note that k01 = and x0 = 1, therefore k02 = Any ecient portfolio is thus the sum of k1 (the minimum variance portfolio) and k2 which...
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## Class Notes in Statistics and Econometrics Part 1 pdf

... Known 10 99 10 99 11 08 11 11 111 6 11 26 11 32 11 46 11 52 11 65 Chapter 54 .1 54.2 54.3 54 Dynamic Linear Models Speciﬁcation and Recursive Solution Locally Constant Model The Reference Model 11 69 11 69 11 75 ... Measures 10 36 10 37 10 38 10 38 10 42 10 43 10 43 10 44 10 45 10 50 Chapter 49 Distributed Lags 49 .1 Geometric lag 49.2 Autoregressive Distributed Lag Models 10 51 1062 10 63 Chapter 50 .1 50.2 50.3 10 73 10 73 10 76 ... Suﬃcient Statistics and Estimation 13 .10 The Likelihood Principle 13 .11 Bayesian Inference Chapter 14 Chapter 15 .1 15.2 15 .3 15 .4 15 .5 15 .6 Interval Estimation 15 Hypothesis Testing Duality between...
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## Class Notes in Statistics and Econometrics Part 2 pptx

... cdf is in (−∞, 2) , 1/36 in [2, 3), 3/36 in [3,4), 6/36 in [4,5), 10/36 in [5,6), 15/36 in [6,7), 21 /36 in [7,8), 26 /36 on [8,9), 30/36 in [9,10), 33/36 in [10,11), 35/36 on [11, 12) , and in [ 12, +∞) ... (1/36,3/36], in (3/36,6/36], in (6/36,10/36], in (10/36,15/36], in (15/36 ,21 /36], in (21 /36 ,26 /36], in (26 /36,30/36], 10 in (30/36,33/36], 11 in (33/36,35/36], and 12 in (35/36,1] 76 RANDOM VARIABLES q ... with and without the notation E[x] = µ: var[x] = E[(x − E[x] )2 ] var[x] = E[(x − µ )2 ] = E[x2 − 2x(E[x]) + (E[x] )2 ] (3.10.17) 2 = E[x ] − 2( E[x]) + (E[x]) = E[x2 ] − (E[x] )2 = E[x2 − 2xµ + µ2...
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## Class Notes in Statistics and Econometrics Part 3 pdf

... in uenced by outliers E[y − 630 0|y≥ 630 0] = E[6000 + 1000z − 630 0|6000 + 1000z≥ 630 0] (5.8. 13) (5.8.14) = E[1000z − 30 0|1000z 30 0] (5.8.15) = E[1000z|z≥0 .3] 30 0 (5.8.16) = 1000 E[z|z≥0 .3] 30 0 ... Pr[2 3 < x 3 ≤ 5 3] = Pr[ 2 3 < x 3 ≤ 5 3 ] = Pr[− < x 3 ≤ 2 2 1 1] = Φ(1) − Φ(− ) = Φ(1) − (1 − Φ( )) = Φ(1) + Φ( ) − = 0.84 13 + 0.6915 − = 0. 532 8 Some tables (Greene) give the area between and ... hiding in the back, and whenever a customer steps up to the counter and rings the bell, they toss a coin If the coin shows head, Herbert serves 154 SPECIFIC RANDOM VARIABLES the customer, and...
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## Class Notes in Statistics and Econometrics Part 4 pps

... 0.5 and min(x, y) ≤ 0 .4] 0. 24 0. 24 = = = Pr[min(x, y) ≤ 0 .4] − 0.36 0. 64 8.3 CONDITIONAL PROBABILITY AND CONDITIONAL MEAN 215 • d Compute the joint cumulative distribution function of u and v Answer ... 227 Problem 140 points Given two independent random variables x and y with density functions fx (x) and gy (y) Write down their joint, marginal, and conditional densities Answer Joint density: ... 136 If the vector random variables x and y have the property that xi is independent of every y j for all i and j, does that make x and y independent random vectors? Interestingly, the answer is...
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## Class Notes in Statistics and Econometrics Part 5 pps

... their values are contained in a linear subspace This has relevance for estimators involving singular random variables: two such estimators (i.e., functions of a singular random variable) should ... r dr, the inner integral becomes − 2π e−t = 2π ; therefore the whole integral is Therefore the product of the integrals of the marginal densities is 1, and since each such marginal integral is ... determined by mean and standard deviation, the bivariate normal is determined by the two means µu and µv , the two standard deviations σu and σv , and the correlation coeﬃcient ρ 10.3.2 Level Lines...
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## Class Notes in Statistics and Econometrics Part 6 pot

... determined by the joint distribution of x and y However the line representing the underlying relationship can only be determined if one has information in addition to the joint density, i.e., in ... permanent income hypothesis, drawing at random families in a given country and asking them about their income y and consumption c can be modeled as the independent observations of two random variables ... underlying line is that it lies between the line predicting y on the basis of x and the line predicting x on the basis of y The name “regression” stems from a confusion between the prediction line and...
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## Class Notes in Statistics and Econometrics Part 7 ppt

... continuous function g which is and nonincreasing for x < and nondecreasing for x > (13.2.3) E[g(|x − θ|)] ≤ E[g(|y − θ|)] 360 13 ESTIMATION PRINCIPLES for every continuous and nondecreasing function ... Cramer Rao 378 13 ESTIMATION PRINCIPLES inequality can also be written in the form (13.4. 27) MSE[t; θ◦ ] ≥ −1/ E◦ [h(θ◦ )] (13.4.26) and (13.4. 27) are usually written in the following form: Assume ... the “indicator function” for nonnegative values, satisfying q(y) = if y ≥ and otherwise We will be working with the random variable which one obtains by inserting the j th observation y j into...
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## Class Notes in Statistics and Econometrics Part 8 ppt

... disparity in hiring rates • b point What would this percentage have to be if training tripled (instead of doubling) one’s chances of getting the job? Answer If training tripled one’s chances of being ... the hiring rates? Answer Since 60% of whites are hired and 40% of the minority group, rm = 60/40 = 1.5 Training is the factor x Sind persons with training had twice the chance of getting the ... the ﬁrms discriminated in hiring? Answer Assuming that the population is in nite, the question is: if one makes 100 independent random drawings from a population that contains 40% minorities, what...
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## Class Notes in Statistics and Econometrics Part 9 pptx

... the vectors Dx2 and Dy? 18.3 THE COEFFICIENT OF DETERMINATION 501 Answer Dx2 is og, the dark blue line starting at the origin, and Dy is cy, the red line starting on x1 and going up to the peak ... is the deterministic part of all the y i , and εi is the random part Model is “simple regression” in which the deterministic part µ is not constant but is a function of the nonrandom variable ... not only in the original 18.2 ORDINARY LEAST SQUARES 491 15-dimensional space, but already in this 3-dimensional subspace In other words, [DM93, Figure 1.3] is valid in all dimensions! In the 15-dimensional...
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## Class Notes in Statistics and Econometrics Part 10 ppt

... strength of a linear association between y and all components of x together, and the partial correlation coeﬃcient measures the strength of that part of the linear association between y and y which ... 229 and 240 To set notation, call the minimum MSE in the ﬁrst prediction (Problem 229) MSE[constant term; y], and the minimum MSE in the second prediction (Problem 240) MSE[constant term and ... ﬁrst and have second highest correlation, and so on Here is the mathematical construction needed: Let x and y be two column vectors consisting of p and q scalar random variables, respectively, and...
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## Class Notes in Statistics and Econometrics Part 11 potx

... to virus attacks and breakins In Linux, the user is the master of the computer and can exploit its full potential • You spend too much time pointing and clicking In GNU/Linux and other unix systems, ... put the following two lines into your batch ﬁles For interactive use on X-terminals you must comment them out again (by putting /* in front and */ after them) 21.5 INSTRUCTIONS FOR STATISTICS 5969, ... domestic two -part tariﬀs in 1935–36, in pence per KWH mc8 The running charge (marginal cost) on domestic two -part tariﬀs in 1937–38, in pence per KWH gas6 The marginal price of gas in 1935–36, in pence...
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## Class Notes in Statistics and Econometrics Part 12 ppsx

... this lefthand side; it is amazing and surprising that it is exactly the population equivalent of the F -test for testing α = in the regression with intercept It can be estimated by replacing α2 ... MSE-matrices are singular, i.e., both estimators allow an error-free look at certain linear combinations of the parameter vector ˆ • b point Give two vectors g = [ g1 ] and h = h1 satisfying MSE[g φ; ... noncentral F has a noncentral χ2 in numerator and a central one in denominator.) The testing principle is therefore: compare the observed value with the upper α point of a F distribution with noncentrality...
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