Lecture 8
Bayesian Inference.
1 Frequentists and Bayesian Paradigms
According to the frequentists theory, it is assumed that unknown parameter is some xed number or
vector. Given parameter value , we observe sample data X from distribution f (|). To estim
Lecture 12
Condence Sets
1 Introduction
So far, we have been considering point estimation. In this lecture, we will study interval estimation. Let
X denote our data. Let R be our parameter of interest. Our task is to construct a data-dependent
interval [l
Name _
AP Chemistry: Nuclear Chemistry
Multiple Choice
214
30. When 84 Po decays, the emission consists consecutively of an alpha particle, then two beta particles, and
finally another alpha particle. The resulting stable nucleus is
210
206
208
210
(A) 20
Name _
AP Chemistry: Nuclear Chemistry
Multiple Choice
30. When 214
84 Po decays, the emission consists consecutively of an alpha particle, then two beta particles, and
finally another alpha particle. The resulting stable nucleus is
206
208
210
210
(A) 20
1. Identical electric charges of +2q are placed at the points
(r,0,0), (0,r,0) and (0,0,r). What is the magnitude of the
electric field at the origin?
3. What is the maximum magnitude of the electric field
possible at point P?
A) 5.0 108 N/C
A) 6q
r2
B) 3
Physics 32.1
Group 5
December 6, 2012
*Capuno, Jodie,
*Lacerna, Zhannis,
*Mallari, Meghan,
* Oclarit, Jamie,
*Policarpio, Gold
*Tolentino, Ahra
MAPPING THE ELECTRIC POTENTIAL AND THE ELECTRIC FIELD
I.
PARALLEL PLATES
1. MEASURE THE POTENTIAL DIFFERENCE BE
Confessions of an College Interviewer
Jay Mathews
2002 Washington Post Newsweek Interactive
2003 Gale Group
For 20 years I volunteered as a Harvard-Radcliffe admissions interviewer. Applicants to the
college, sometimes as many as a dozen in a year, came
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Lecture 7
Maximum Likelihood Estimation.
1
Attainability of Rao-Cramer bound
In the last lecture we derived the Rao-Cramer bound such that no unbiased estimator may have lower variance
than this bound, at least under some regularity conditions. Now we can
Lecture 9
Testing Concepts.
1 Hypotheses
Hypotheses are some statements about population distribution, which are either true or untrue for the given
population.
Example For example, let X1 , ., Xn be a random sample from distribution N (, 2 ) with 2 known
V.C. 14.381 Class Notes1
1. Fundamentals of Regression
1.1. Regression and Conditional Expectation Function. Suppose yt is a real
response variable, and wt is a d-vector of covariates. We are interested in the condi
tional mean (expectation) of yt given w
14.381 Practice Final
Fall 2005
Question 1. Relaxing assumptions on WLLN (Theorem 5.5.2).
n
a) Show that p lim X
= for the same assumptions on Xi as in Theorem 5.5.2 except
that now we replace the independence assumption with the assumption that the Xi a
14.381, Fall 2006
Problem Set 5
Due: Nov. 16, 2006 (in class)
1. Derive the OLS estimate for . So much of what fol lows builds o of the basic
ideas contained in this solution that you should be able to construct in your sleep.
Yes, you can just copy the
Lecture 4
Sucient Statistics. Factorization Theorem
1 Sucient statistics
Let f (x|) with be some parametric family. Let X = (X1 , ., Xn ) be a random sample from distribution
f (x|). Suppose we would like to learn parameter value from our sample. The conc
Lecture 5
Point estimators.
1 Estimators. Properties of estimators.
An estimator is a function of the data. If we have a parametric family with parameter , then an estimator
of is usually denoted by .
Example For example, if X1 , ., Xn is a random sample
Lecture 3
Intro to Statistics. Inferences for normal families.
1 Random Sampling
Often, the data collected in an experiment consist of several observations of a variable of interest. This
situation is often described by a model called random sampling.
Den
Lecture 2
Convergence theorems
1 Types of Convergence
To deal with convergence arguments in the next section, we have to introduce the concept of a random
experiment. By denition, a random experiment is a triple (, A, P ) where denotes the sample space,
A
Lecture 1
Distributions and Normal Random Variables
1 Random variables
1.1 Basic Denitions
Given a random variable X , we dene a cumulative distribution function (cdf ), FX : R [0, 1], such that
FX (t) = P cfw_X t for all t R. Here P cfw_X t denotes the p
Lecture 6
Ecient estimators. Rao-Cramer bound.
1 MSE and Suciency
Let X = (X1 , ., Xn ) be a random sample from distribution f . Let = (X) be an estimator of . Let
T (X) be a sucient statistic for . As we have seen already, MSE provides one way to compare
Lecture 10
Uniformly Most Powerful Tests.
1 Uniformly Most Powerful Test
Let = 0 1 be a parameter space. Consider a parametric family cfw_f (x|), . Suppose we want to
test the null hypothesis, H0 , that 0 against the alternative, Ha , that 1 . Let C be so
Lecture 11
Large Sample Tests.
1 Likelihood Ratio Test
Let X1 , ., Xn be a random sample from a distribution with pdf f (x|) where is some one dimensional
(unknown) parameter. Suppose we want to test the null hypothesis, H0 , that = 0 against the alternat
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