IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty  Observation mechanism:  Priors:  Parameters Role in inverse.

Slides:

Advertisements

Similar presentations

TWO STEP EQUATIONS 1. SOLVE FOR X 2. DO THE ADDITION STEP FIRST

Advertisements

You have been given a mission and a code. Use the code to complete the mission and you will save the world from obliteration…

Chapter 5 One- and Two-Sample Estimation Problems.

Fundamentals of Probability

Advanced Piloting Cruise Plot.

Chapter 1 The Study of Body Function Image PowerPoint

Copyright © 2011, Elsevier Inc. All rights reserved. Chapter 5 Author: Julia Richards and R. Scott Hawley.

1 Copyright © 2010, Elsevier Inc. All rights Reserved Fig 2.1 Chapter 2.

STATISTICS Joint and Conditional Distributions

STATISTICS Sampling and Sampling Distributions

STATISTICS HYPOTHESES TEST (II) One-sample tests on the mean and variance Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National.

STATISTICS Random Variables and Distribution Functions

By D. Fisher Geometric Transformations. Reflection, Rotation, or Translation 1.

Properties Use, share, or modify this drill on mathematic properties. There is too much material for a single class, so you’ll have to select for your.

Business Transaction Management Software for Application Coordination 1 Business Processes and Coordination.

Jeopardy Q 1 Q 6 Q 11 Q 16 Q 21 Q 2 Q 7 Q 12 Q 17 Q 22 Q 3 Q 8 Q 13

Jeopardy Q 1 Q 6 Q 11 Q 16 Q 21 Q 2 Q 7 Q 12 Q 17 Q 22 Q 3 Q 8 Q 13

Title Subtitle.

My Alphabet Book abcdefghijklm nopqrstuvwxyz.

DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

MULTIPLYING MONOMIALS TIMES POLYNOMIALS (DISTRIBUTIVE PROPERTY)

SUBTRACTING INTEGERS 1. CHANGE THE SUBTRACTION SIGN TO ADDITION

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

FACTORING Think Distributive property backwards Work down, Show all steps ax + ay = a(x + y)

FACTORING ax2 + bx + c Think “unfoil” Work down, Show all steps.

Year 6 mental test 5 second questions

Probability Distributions

STATISTICAL INFERENCE ABOUT MEANS AND PROPORTIONS WITH TWO POPULATIONS

Solve Multi-step Equations

Richmond House, Liverpool (1) 26 th January 2004.

REVIEW: Arthropod ID. 1. Name the subphylum. 2. Name the subphylum. 3. Name the order.

BT Wholesale October Creating your own telephone network WHOLESALE CALLS LINE ASSOCIATED.

Tutorial 8, STAT1301 Fall 2010, 16NOV2010, By Joseph Dong.

ABC Technology Project

5-1 Chapter 5 Theory & Problems of Probability & Statistics Murray R. Spiegel Sampling Theory.

1 Undirected Breadth First Search F A BCG DE H 2 F A BCG DE H Queue: A get Undiscovered Fringe Finished Active 0 distance from A visit(A)

1 Breadth First Search s s Undiscovered Discovered Finished Queue: s Top of queue 2 1 Shortest path from s.

“Start-to-End” Simulations Imaging of Single Molecules at the European XFEL Igor Zagorodnov S2E Meeting DESY 10. February 2014.

IPIM, IST, José Bioucas, Convolution Operators Spectral Representation Bandlimited Signals/Systems Inverse Operator Null and Range Spaces Sampling,

Factor P 16 8(8-5ab) 4(d² + 4) 3rs(2r – s) 15cd(1 + 2cd) 8(4a² + 3b²)

Squares and Square Root WALK. Solve each problem REVIEW:

Do you have the Maths Factor?. Maths Can you beat this term’s Maths Challenge?

The Normal Distribution PSYC 6130, PROF. J. ELDER 2 is the mean is the standard deviation The height of a normal density curve at any point x is given.

© 2012 National Heart Foundation of Australia. Slide 2.

Lets play bingo!!. Calculate: MEAN Calculate: MEDIAN

Understanding Generalist Practice, 5e, Kirst-Ashman/Hull

Chapter 5 Test Review Sections 5-1 through 5-4.

SIMOCODE-DP Software.

GG Consulting, LLC I-SUITE. Source: TEA SHARS Frequently asked questions 2.

Addition 1’s to 20.

25 seconds left…...

Januar MDMDFSSMDMDFSSS

We will resume in: 25 Minutes.

©Brooks/Cole, 2001 Chapter 12 Derived Types-- Enumerated, Structure and Union.

A SMALL TRUTH TO MAKE LIFE 100%

1 Unit 1 Kinematics Chapter 1 Day

PSSA Preparation.

How Cells Obtain Energy from Food

CpSc 3220 Designing a Database

Traktor- og motorlære Kapitel 1 1 Kopiering forbudt.

DISTRIBUSI PROBABILITAS KONTINYU Referensi : Walpole, RonaldWalpole. R.E., Myers, R.H., Myers, S.L., and Ye, K Probability & Statistics for Engineers.

Presentation transcript:

IP, IST, José Bioucas, Probability The mathematical language to quantify uncertainty  Observation mechanism:  Priors:  Parameters Role in inverse problems

IP, IST, José Bioucas, Overview of Probability  Definition; Properties  Independency; Conditional probability; Bayes theorem  Random variables; Cumulative distribution function  Examples of random variables  Bivariate distributions; Marginal distribution;Conditional distribution  Multivariate distributions; Marginal distribution;Conditional distributions  Expectation of a random variable; Variance; Covariance  Conditional expectation of a random variable; Variance; Covariance  Weak law of large numbers Ref. Larry Wasserman, All of Statistics. A Concise Course in Statistical Inference, Springer, 2004

IP, IST, José Bioucas, Definition Frequencist interpretation: Number of occurencies of A Number of repetitions Bayesain interpretation: Measures an observer’s strength of belief that A is true In probability the interpretation does not matter sample space event

IP, IST, José Bioucas, Independency; Conditional probability    

IP, IST, José Bioucas, Random variables  

IP, IST, José Bioucas, Random variables   

IP, IST, José Bioucas, Some discrete random variables From Wikipedia

IP, IST, José Bioucas, Some discrete random variables From Wikipedia

IP, IST, José Bioucas, Some continuous random variables

IP, IST, José Bioucas, Some continuous random variables

IP, IST, José Bioucas, Some continuous random variables

IP, IST, José Bioucas, Bivariate distributions

IP, IST, José Bioucas, Bivariate distributions

IP, IST, José Bioucas, Bivariate distributions

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Expectation

IP, IST, José Bioucas, Multivariate Normal

IP, IST, José Bioucas, Multivariate Normal

IP, IST, José Bioucas, Inequalities

IP, IST, José Bioucas, Laws of large numbers

IP, IST, José Bioucas, Central Limit Theorem