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

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

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

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

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

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

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

Topic 15: Maximum Likelihood Estimation

Introduction to Statistical Methodology Maximum Likelihood Estimation Exercise 3. Check that this is a maximum. Thus, p^(x) = x: In this case the maximum likelihood estimator is also unbiased. Example 4 (Normal data). Maximum likelihood estimation can be applied to a vector valued parameter. For a simple

  Topics, Maximum, Estimation, Likelihood, Maximum likelihood estimation, Topic 15

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

CONDITIONAL PROBABILITY Discrete random variables ...

By: PNeil E. Cotter ROBABILITY CONDITIONAL PROBABILITY Discrete random variables DEFINITIONS AND FORMULAS DEF: P(A|B) ≡ the (conditional) Probability of A given B occurs NOT'N: | ≡ "given" EX: The probability that event A occurs may change if we know event B has occurred. For example, if A ≡ it will snow today, and if B ≡ it is 90° outside, then knowing that

  Variable, Probability, Random, Random variables

Reading 5b: Continuous Random Variables

Continuous Random Variables and Probability Density Func­ tions. A continuous random variable takes a range of values, which may be finite or infinite in extent. Here are a few examples of ranges: [0, 1], [0, ∞), (−∞, ∞), [a, b]. Definition: A random variable X is continuous if there is a function f(x) such that for any c ≤ d we ...

  Variable, Probability, Random, Random variables

Chapter 4 RANDOM VARIABLES - University of Kent

CONTINUOUS RANDOM VARIABLES Introduction Reminder: a rv is said to be continuous if its cdf is a continuous function. If the function FX(x) = Pr(X ≤ x) of x is continuous, what is Pr(X = x)? Pr(X = x) = Pr(X ≤ x) − Pr(X < x) = 0, by continuity A continuous random variable does not possess a probability function.

  Variable, Probability, Random, Random variables

Random Variables, Distributions, and Expected Value

Random Variables, Distributions, and Expected Value Fall2001 ProfessorPaulGlasserman B6014: ManagerialStatistics 403UrisHall The Idea of a Random Variable 1. A random variable is a variable that takes specific values with specific probabilities. ... Right panel shows a probability density for a continuous random variable. The probabilityP ...

  Distribution, Value, Expected, Variable, Probability, Random, Random variables, And expected value

Random Variables and Probability Distributions

A Random Variables and Probability Distributions A.1 Distribution Functions and Expectation A.2 Random Vectors A.3 The Multivariate Normal Distribution A.1 Distribution Functions and Expectation The distribution function F of a random variable X is defined by F(x) = P[X ≤ x] (A.1.1) for all real x. The following properties are direct ...

  Variable, Probability, Random, Random variables

Probability, Statistics, and Random Processes for ...

vi Contents CHAPTER 4 One Random Variable 141 4.1 The Cumulative Distribution Function 141 4.2 The Probability Density Function 148 4.3 The Expected Value of X 155 4.4 Important Continuous Random Variables 163

  Variable, Probability, Random, Random variables

Reading 4b: Discrete Random Variables: Expected Value

class 4, Discrete Random Variables: Expected Value, Spring 2014 4 It is possible to show that the sum of this series is indeed np. We think you’ll agree that the method using Property (1) is much easier. Example 8. (For infinite random variables …

  Variable, Random, Random variables

Probability, Random Processes, and Ergodic Properties

Random processes with standard alphabets We develop the theory of standard spaces as a model of quite general process alphabets. Although not as general (or abstract) as examples often considered by probability theorists, standard spaces have useful structural properties

  Probability, Random, Ergodic