Topic 15: Maximum Likelihood Estimation

Joint and Marginal Distributions - University of Arizona

Joint and Marginal Distributions October 23, 2008 We will now consider more than one random variable at a time. As we shall see, developing the theory of multivariate distributions will allow us to consider situations that model the actual collection of data and form the foundation of inference based on those data. 1 Discrete Random Variables

  Marginal

Chapter 6 Importance sampling

random variable we want to compute the mean of is of the form f(X~) where X~ is a random vector. We will assume that the joint distribution of X~ is absolutely continous and let p(~x) be the density. (Everything we will do also works for the case where the random vector X~ is discrete.) So we focus on computing Ef(X~) = Z f(~x)p(~x)dx (6.1)

  Sampling, Random

1 Sufficient statistics

conditional distribution. But then his random sample has the same distri-bution as a random sample drawn from the population (with its unknown value of θ). So statistician B can use his random sample X0 1,···,X0 n to com-pute whatever statistician A computes using his random sample X1,···,Xn, and he will (on average) do as well as ...

  Random, Conditional

A Conditional expectation

The partition theorem says that if Bn is a partition of the sample space then E[X] = X n E[XjBn]P(Bn) Now suppose that X and Y are discrete RV’s. If y is in the range of Y then Y = y is a event with nonzero probability, so we can use it as the B in the above.

Method of Moments - University of Arizona

The muon is an elementary particle with an electric charge of 1 and a spin (an intrinsic angular momentum) of 1/2. It is an unstable subatomic particle with a mean lifetime of 2.2 µs. Muons have a mass of about 200 times the mass of an electron. Since the muon’s charge and spin are the same as the electron, a muon can be

  Methods, Moment, Muon, Method of moments

Topic 15 Maximum Likelihood Estimation

Maximum Likelihood Estimation Multidimensional Estimation 1/10. Fisher Information Example Outline Fisher Information Example Distribution of Fitness E ects ... To obtain the maximum likelihood estimate for the gamma family of random variables, write the likelihood L( ; jx) = ( ) x 1 1 e x1 ( ) x 1 n e xn = ( ) n (x 1x 2 x

  Maximum, Estimation, Likelihood, Maximum likelihood estimation, Maximum likelihood

Maximum Likelihood Estimation - University of Arizona

Introduction to the Science of Statistics Maximum Likelihood Estimation 0.2 0.3 0.4 0.5 0.6 0.7 0.0e+00 5.0e-07 1.0e-06 1.5e-06 p l 0.2 0.3 0.4 0.5 0.6 0.7

  Estimation

Innovative Methods of Teaching - University of Arizona

The reason being that martyr is engaged in defense work while an alim (scholar) builds individuals and nations along positive lines. In this way he bestows a real life to the world. “Education is the manifestation of perfection already in man” – (Swami Vivekananda)

  Martyr

Probability Theory - University of Arizona

Probability Theory December 12, 2006 Contents 1 Probability Measures, Random Variables, and Expectation 3 ... Definition 1.18. Let f : (S,S) → (T,T ) be a function between measure spaces, then f is called measurable if f−1(B) ∈ S for every B ∈ T . (1.6) If (S,S) has a probability measure, then f is called a random variable.

  Measure, Theory, Space, Measurable, Measure spaces

Interval Estimation - University of Arizona

likelihood, and evaluate the quality of the estimator by evaluating the bias and the variance of the estimator. Often, we know more about the distribution of the estimator and this allows us to take a more comprehensive statement about the estimation procedure. Interval estimation is an alternative to the variety of techniques we have examined.

  Estimation, Likelihood

Logarithm Formulas - Austin Mohr

Logarithm Formulas Expansion/Contraction Properties of Logarithms These rules are used to write a single complicated logarithm as several simpler logarithms (called \ex-panding") or several simple logarithms as a single complicated logarithm (called \contracting"). Notice that these rules work for any base. log a (xy) = log a (x) + log a

  Formula, Algorithm, Logarithm formulas

An Introduction to the Theory of Elliptic Curves

An Introduction to the Theory of Elliptic Curves The Discrete Logarithm Problem Fix a group G and an element g 2 G.The Discrete Logarithm Problem (DLP) for G is: Given an element h in the subgroup generated by g, flnd an integer m satisfying h = gm: The smallest integer m satisfying h = gm is called the logarithm (or index) of h with respect to g, and is denoted

  Introduction, Theory, Algorithm, Curves, Elliptic, An introduction to the theory of elliptic curves

Common Derivatives Integrals - Lamar University

Exponential/Logarithm Functions (xx) ln() d aaa dx = ( ) d dx ee= (ln())1,0 d xx dxx => 1 ln,0 d xx dxx =„ (()) 1 log,0 a ln d xx dxxa => Hyperbolic Trig Functions (sinh) cosh d xx dx = (cosh) sinh d xx dx = (tanh) sech2 d xx dx = (sech) sechtanh d xxx dx =-(csch) cschcoth d xxx dx =-(coth) csch2 d xx dx =-Common Derivatives and Integrals ...

  University, Lamar university, Lamar, Algorithm

Algebra Cheat Sheet - Pauls Online Math Notes

For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins Logarithms and Log Properties Definition log is equivalent to y ...

  Math, Tutorials, Lamar

CALCULUS I - hi

Differentiation Formulas – Here we will start introducing some of the differentiation formulas used in a calculus course. Product and Quotient Rule – In this section we will took at differentiating products and quotients of functions. Derivatives of Trig Functions – We’ll give the derivatives of the trig functions in this section.

  Formula

Maximum Likelihood Estimation - University of Arizona

Figure 15.1: Likelihood function (top row) and its logarithm (bottom row) for Bernouli trials. The left column is based on 20 trials having 8 and 11 successes. The right column is based on 40 trials having 16 and 22 successes. Notice that the maximum likelihood is approximately 10 6 for 20 trials and 10 12 for 40. In addition, note that the ...

  Algorithm

Math Handbook of Formulas, Processes and Tricks

109 Exponent Formulas 110 Logarithm Formulas 111 e 112 Table of Exponents and Logs 113 Converting Between Exponential and Logarithmic Forms 114 Expanding Logarithmic Expressions 115 Condensing Logarithmic Expressions 116 Condensing Logarithmic Expressions – More Examples 117 Graphing an Exponential Function

  Formula, Math, Algorithm, Logarithm formulas

Properties of Logarithms Worksheet - VealeyMath

I) Model Problems For any positive numbers X, Y and N and any positive base b, the following formulas are true: log b X N = N • log b X Power Rule for Logarithms log b X Y § ©¨ · ¹¸ = log b X – log b Y Quotient Rule for Logarithms log b (XY) = logb X + log b Y Product Rule for Logarithms The following examples show how to expand logarithmic expressions using each of the

  Formula, Properties, Algorithm, Properties of logarithms