Derivations of Student’s-T and the F Distributions

Slides:

Advertisements

Similar presentations

STATISTICS Joint and Conditional Distributions

Advertisements

Functions of Random Variables. Method of Distribution Functions X 1,…,X n ~ f(x 1,…,x n ) U=g(X 1,…,X n ) – Want to obtain f U (u) Find values in (x 1,…,x.

INTRODUCTION TO COPULAS

Bivariate Normal Distribution and Regression Application to Galton’s Heights of Adult Children and Parents Sources: Galton, Francis (1889). Natural Inheritance,

Chapter 2 Multivariate Distributions Math 6203 Fall 2009 Instructor: Ayona Chatterjee.

Bayesian inference of normal distribution

Joint and marginal distribution functions For any two random variables X and Y defined on the same sample space, the joint c.d.f. is For an example, see.

1. (a) (b) The random variables X 1 and X 2 are independent, and each has p.m.f.f(x) = (x + 2) / 6 if x = –1, 0, 1. Find E(X 1 + X 2 ). E(X 1 ) = E(X 2.

CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics Slides by Michael Maurizi Instructor Longin Jan Latecki C9:

Probability Theory STAT 312 STAT 312 Dr. Zakeia AlSaiary.

Probability theory 2010 Main topics in the course on probability theory  Multivariate random variables  Conditional distributions  Transforms  Order.

1 Engineering Computation Part 6. 2 Probability density function.

Probability theory 2011 Main topics in the course on probability theory  The concept of probability – Repetition of basic skills  Multivariate random.

Composite Beam Transformed homogeneous beam obtained through a transformation factor: n = E1E2E1E2 dF = σ dA = σ dA’ σ dz dy = σ’ n dz dy σ = n σ’ and.

Chapter 3. Transport Equations The plasma flows in planetary ionospheres can be in equilibrium (d/dt = 0), like the midlatitude terrestrial ionosphere,

4.7 Brownian Bridge(part 2) 報告人：李振綱. Outline Multidimensional Distribution of the Brownian Bridge Multidimensional Distribution of the Brownian.

Probability theory 2010 Conditional distributions  Conditional probability:  Conditional probability mass function: Discrete case  Conditional probability.

Continuous Random Variables and Probability Distributions

Multivariate Distributions. Distributions The joint distribution of two random variables is f(x 1,x 2 ) The marginal distribution of f(x 1 ) is obtained.

Chapter 4 Joint Distribution & Function of rV. Joint Discrete Distribution Definition.

5-1 Two Discrete Random Variables Example Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

5-1 Two Discrete Random Variables Example Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

KINEMATICS ANALYSIS OF ROBOTS (Part 1) ENG4406 ROBOTICS AND MACHINE VISION PART 2 LECTURE 8.

Distribution Function properties. Density Function – We define the derivative of the distribution function F X (x) as the probability density function.

Pairs of Random Variables Random Process. Introduction  In this lecture you will study:  Joint pmf, cdf, and pdf  Joint moments  The degree of “correlation”

Today we will derive and use the formula for the area of a parallelogram by comparing it with the formula for the area of a rectangle. derive = obtain.

CHAPTER Continuity Integration by Parts The formula for integration by parts  f (x) g’(x) dx = f (x) g(x) -  g(x) f’(x) dx. Substitution Rule that.

PBG 650 Advanced Plant Breeding

PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Two Functions of Two Random.

1 7. Two Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to.

Chapter 3 Random vectors and their numerical characteristics.

Chapter 8 Multivariable Calculus Section 2 Partial Derivatives.

Energy momentum tensor of macroscopic bodies Section 35.

Bayesian statistics Probabilities for everything.

Multiple Random Variables Two Discrete Random Variables –Joint pmf –Marginal pmf Two Continuous Random Variables –Joint Distribution (PDF) –Joint Density.

1 Two Functions of Two Random Variables In the spirit of the previous lecture, let us look at an immediate generalization: Suppose X and Y are two random.

The forces on a conductor Section 5. A conductor in an electric field experiences forces.

PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Two Random Variables.

Chapter 5a:Functions of Random Variables Yang Zhenlin.

Chapter 3 Differential Motions and Velocities

Distribution Function properties Let us consider the experiment of tossing the coin 3 times and observing the number of heads The probabilities and the.

Continuous Random Variables and Probability Distributions

EEE APPENDIX B Transformation of RV Huseyin Bilgekul EEE 461 Communication Systems II Department of Electrical and Electronic Engineering Eastern.

Tables 1 to 8 FM Run around!.

Chapter 5 Joint Probability Distributions and Random Samples  Jointly Distributed Random Variables.2 - Expected Values, Covariance, and Correlation.3.

I NTEGRATION BY P ARTS Section 8-2. Integration by Parts: undo the product rule 1. The derivative of the product of two functions u and v “First times.

6 vector RVs. 6-1: probability distribution A radio transmitter sends a signal to a receiver using three paths. Let X1, X2, and X3 be the signals that.

Chapter 9: Joint distributions and independence CIS 3033.

Statistics Lecture 19.

Ch4.2 Cumulative Distribution Functions

Main topics in the course on probability theory

PRODUCT MOMENTS OF BIVARIATE RANDOM VARIABLES

Conditional Probability on a joint discrete distribution

Some Rules for Expectation

Integration The Explanation of integration techniques.

Integration The Explanation of integration techniques.

Question Suppose exists, find the limit: (1) (2) Sol. (1) (2)

Copyright © Cengage Learning. All rights reserved.

7. Two Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record.

Preliminaries: Distributions

ASV Chapters 1 - Sample Spaces and Probabilities

South Dakota School of Mines & Technology Expectations for Exponential

CS723 - Probability and Stochastic Processes

7. Two Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record.

9. Two Functions of Two Random Variables

Concept 7: The Second Fundamental Theorem of Calculus

7. Two Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record.

Introduction to Probability: Solutions for Quizzes 4 and 5

Berlin Chen Department of Computer Science & Information Engineering

Presentation transcript:

Derivations of Student’s-T and the F Distributions

Student’s-T Distribution (P. 1)

Student’s T-Distribution (P. 2) Step 1: Fix V=v and write f(z|v)=f(z) (by independence) Step 2: Let T = h(Z) (and Z=h-1(T)) and obtain f(t|v) by method of transformations: fT(t|v) = fZ(h-1(t)|v)|dZ/dT| Step 3: Obtain joint distribution of T, V : fT,V(t,v) = fT(t|v) fV(v) Step 4: Obtain marginal distribution of T by integrating the joint density over V and putting in form:

Student’s-T Distribution (P. 3) Conditional Distribution of T|V=v and Marginal Distribution of V

Student’s-T Distribution (P. 4) Marginal Distribution of T (integrating out V) (Continued below)

Student’s-T Distribution (P. 5) Marginal Distrbution of T

F-Distribution (P. 1)

F-Distribution (P.2) Step 1: Fix W=w f(v|w) = f(v) (independence) Step 2: Let F=h(V) and V=h-1(F) and obtain fF(f|w) by method of transformations: fF(f|w) = fV(h-1(f)|w) |dV/dF| Step 3: Obtain the joint distribution of F and W fF,W(f,w) = fF(f|w) fW(w) Step 4: Obtain marginal distribution of F by integrating joint density over W and putting in form:

Conditional Distribution of F|W=w F-Distribution (P. 3) Conditional Distribution of F|W=w

Marginal Distribution of W, Joint Distribution of F,W F-Distribution (P. 4) Marginal Distribution of W, Joint Distribution of F,W

Marginal Distribution of F F-Distribution (P. 5) Marginal Distribution of F