Probability Review Definitions/Identities Random Variables

Slides:

Advertisements

Similar presentations

Probability Learning Outcomes  I can find the probability of simple events, combined events and use expectation  I can use the addition rule for Mutually.

Advertisements

Intro to Bayesian Learning Exercise Solutions Ata Kaban The University of Birmingham 2005.

Probability Notation Review Prior (unconditional) probability is before evidence is obtained, after is posterior or conditional probability P(A) – Prior.

Lesson #9 Rules of Probability. Venn Diagram S A.

1 Probability. 2 Probability has three related “meanings.” 1. Probability is a mathematical construct. Probability.

Mutually Exclusive: P(not A) = 1- P(A) Complement Rule: P(A and B) = 0 P(A or B) = P(A) + P(B) - P(A and B) General Addition Rule: Conditional Probability:

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.

Chapter 6 Probabilit y Vocabulary Probability – the proportion of times the outcome would occur in a very long series of repetitions (likelihood of an.

1 CY1B2 Statistics Aims: To introduce basic statistics. Outcomes: To understand some fundamental concepts in statistics, and be able to apply some probability.

5.3B Conditional Probability and Independence Multiplication Rule for Independent Events AP Statistics.

A.P. STATISTICS LESSON 6.3 ( DAY 2 ) GENERAL PROBABILITY RULES ( EXTENDED MULTIPLICATION RULES )

26134 Business Statistics Maths Study Centre CB Tutorial 8: Probability Distribution Key concepts in this tutorial.

AP Statistics Chapter 6 Notes. Probability Terms Random: Individual outcomes are uncertain, but there is a predictable distribution of outcomes in the.

Probability. Statistical inference is based on a Mathematics branch called probability theory. If a procedure can result in n equally likely outcomes,

Tree Diagram Worksheet

Dr. Omar Al Jadaan Probability. Simple Probability Possibilities and Outcomes Expressed in the form of a fraction A/B Where A is the occurrence B is possible.

AP STATISTICS LESSON 6.3 (DAY 1) GENERAL PROBABILITY RULES.

Appendix : Probability Theory Review Each outcome is a sample point. The collection of sample points is the sample space, S. Sample points can be aggregated.

Dr. Ahmed Abdelwahab Introduction for EE420. Probability Theory Probability theory is rooted in phenomena that can be modeled by an experiment with an.

9/26 디지털 영상통신 Mathematical Preliminaries Math Background Predictive Coding Huffman Coding Matrix Computation.

Section 3.2 Notes Conditional Probability. Conditional probability is the probability of an event occurring, given that another event has already occurred.

NLP. Introduction to NLP Very important for language processing Example in speech recognition: –“recognize speech” vs “wreck a nice beach” Example in.

Probability You’ll probably like it!. Probability Definitions Probability assignment Complement, union, intersection of events Conditional probability.

1 Probability: Liklihood of occurrence; we know the population, and we predict the outcome or the sample. Statistics: We observe the sample and use the.

8-3: Probability and Probability Distributions English Casbarro Unit 8.

4.3 More Discrete Probability Distributions NOTES Coach Bridges.

1 Probability: Introduction Definitions,Definitions, Laws of ProbabilityLaws of Probability Random VariablesRandom Variables DistributionsDistributions.

Probability. Probability Probability is fundamental to scientific inference Probability is fundamental to scientific inference Deterministic vs. Probabilistic.

Probability. Randomness When we produce data by randomized procedures, the laws of probability answer the question, “What would happen if we did this.

Central Limit Theorem Let X 1, X 2, …, X n be n independent, identically distributed random variables with mean  and standard deviation . For large n:

STATISTICS 6.0 Conditional Probabilities “Conditional Probabilities”

BUSA Probability. Probability – the bedrock of randomness Definitions Random experiment – observing the close of the NYSE and the Nasdaq Sample.

Hidden Markov Models. A Hidden Markov Model consists of 1.A sequence of states {X t |t  T } = {X 1, X 2,..., X T }, and 2.A sequence of observations.

CHAPTER 12 General Rules of Probability BPS - 5TH ED.CHAPTER 12 1.

Probability – the bedrock of randomness Definitions Random experiment – observing the close of the NYSE and the Nasdaq Sample space = {NYSE+Nasdaq+, NYSE+Nasdaq-,

Section 3.3 Addition Rule (Venn Diagram). Section 3.3 Objectives Determine if two events are mutually exclusive Use a Venn Diagram to find the probability.

Conditional Probability 423/what-is-your-favorite-data-analysis-cartoon 1.

© 2005 McGraw-Hill Ryerson Ltd. 4-1 Statistics A First Course Donald H. Sanders Robert K. Smidt Aminmohamed Adatia Glenn A. Larson.

Basics of Multivariate Probability

MECH 373 Instrumentation and Measurements

Probability and Statistics

Chapter 3 Probability Slides for Optional Sections

Chapter 3 Probability.

Quick Review Probability Theory

Tutorial 8: Probability Distribution

From Randomness to Probability

Tutorial 8: Probability Distribution

An Example of {AND, OR, Given that} Using a Normal Distribution

Lecture 11 Sections 5.1 – 5.2 Objectives: Probability

Hidden Markov Autoregressive Models

Advanced Artificial Intelligence

The Binomial Distribution

Welcome to the wonderful world of Probability

10-3 Probability distributions

... DISCRETE random variables X, Y Joint Probability Mass Function y1

How to find probabilities of 2 events using a tree diagram

ASV Chapters 1 - Sample Spaces and Probabilities

Introduction to Probability

ASV Chapters 1 - Sample Spaces and Probabilities

CS723 - Probability and Stochastic Processes

General Probability Rules

More About Probability

Probability Multiplication law for dependent events

Review: Mini-Quiz Combined Events

Chapter 3 & 4 Notes.

Presentation transcript:

Probability Review Definitions/Identities Random Variables Expected Value Joint Distributions Conditional Probabilities

Probability Defined an event (experiment) has a set of possible outcomes, each with a probability, that measures their relative (anticipated) frequencies of occurrence normalized to 1.

Probability Identities Events and outcomes: Probability of each outcome: Probability distribution:

Joint Distributions Two (or more) events Each event has an outcome Joint distribution stipulates the probability of every combination of outcomes

Two Events

Random Variables

Multiple Random Variables

Joint probability matrix

Conditional Probability Random variables are often NOT independent P(rain in Pittsburgh), P(rain in Monroeville), P(rain in New York), P(rain in Hong Kong) P(Heads up), P(Tails down) P(D1=5), P(D2=6) P(D1=1), P(D1 + D2=2)

Dice Example

Conditional Probability P(A|B) = P(AB) P(B) AB

Example p(y1) = 0.2 p(y2) = 0.1 p(y3) = 0.7

Markov Processes State transition probabilities Matrix or Diagram Matrix Multiplication predicts multiple transition probabilities Mk Converges to steady state