Probability Refresher COMP5416 Advanced Network Technologies.

Slides:

Advertisements

Similar presentations

Data Analysis Class 4: Probability distributions and densities.

Advertisements

The Normal Distribution

Special random variables Chapter 5 Some discrete or continuous probability distributions.

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law.

1 Continuous random variables Continuous random variable Let X be such a random variable Takes on values in the real space  (-infinity; +infinity)  (lower.

Discrete Uniform Distribution

EXAMPLE 1 Construct a probability distribution

EXAMPLE 1 Construct a probability distribution Let X be a random variable that represents the sum when two six-sided dice are rolled. Make a table and.

Some Continuous Probability Distributions By: Prof. Gevelyn B. Itao.

Review of Basic Probability and Statistics

Continuous Random Variables. For discrete random variables, we required that Y was limited to a finite (or countably infinite) set of values. Now, for.

Descriptive statistics Experiment  Data  Sample Statistics Sample mean Sample variance Normalize sample variance by N-1 Standard deviation goes as square-root.

MA 102 Statistical Controversies Monday, April 1, 2002 Today: Randomness and probability Probability models and rules Reading (for Wednesday): Chapter.

TDC 369 / TDC 432 April 2, 2003 Greg Brewster. Topics Math Review Probability –Distributions –Random Variables –Expected Values.

1 Review of Probability Theory [Source: Stanford University]

Descriptive statistics Experiment  Data  Sample Statistics Experiment  Data  Sample Statistics Sample mean Sample mean Sample variance Sample variance.

Engineering Probability and Statistics - SE-205 -Chap 3 By S. O. Duffuaa.

A random variable that has the following pmf is said to be a binomial random variable with parameters n, p The Binomial random variable.

2. Random variables  Introduction  Distribution of a random variable  Distribution function properties  Discrete random variables  Point mass  Discrete.

The moment generating function of random variable X is given by Moment generating function.

Discrete Random Variables and Probability Distributions

Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint.

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

CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics, 2007 Instructor Longin Jan Latecki Chapter 7: Expectation and variance.

B AD 6243: Applied Univariate Statistics Understanding Data and Data Distributions Professor Laku Chidambaram Price College of Business University of Oklahoma.

Week 41 Continuous Probability Spaces Ω is not countable. Outcomes can be any real number or part of an interval of R, e.g. heights, weights and lifetimes.

Chapter 5 Statistical Models in Simulation

Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Random Variables.

Random Variables & Probability Distributions Outcomes of experiments are, in part, random E.g. Let X 7 be the gender of the 7 th randomly selected student.

Exponential Distribution

Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution.

Random Variables and Stochastic Processes –

Random Variables. A random variable X is a real valued function defined on the sample space, X : S  R. The set { s  S : X ( s )  [ a, b ] is an event}.

Probability Theory Overview and Analysis of Randomized Algorithms Prepared by John Reif, Ph.D. Analysis of Algorithms.

MAT 1235 Calculus II Section 8.5 Probability

STA347 - week 31 Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5’s in the 6 rolls. Let X = number of.

Continuous Probability Distributions Statistics for Management and Economics Chapter 8.

Methodology Solving problems with known distributions 1.

Starting point for generating other distributions.

IE 300, Fall 2012 Richard Sowers IESE. 8/30/2012 Goals: Rules of Probability Counting Equally likely Some examples.

School of Information Technologies Poisson-1 The Poisson Process Poisson process, rate parameter e.g. packets/second Three equivalent viewpoints of the.

Lec. 08 – Discrete (and Continuous) Probability Distributions.

Review of Chapter

4.3 More Discrete Probability Distributions NOTES Coach Bridges.

Psychology 202a Advanced Psychological Statistics September 22, 2015.

Random Variables Example:

3.1 Statistical Distributions. Random Variable Observation = Variable Outcome = Random Variable Examples: – Weight/Size of animals – Animal surveys: detection.

MAT 1235 Calculus II Section 8.5 Probability

1 Opinionated in Statistics by Bill Press Lessons #15.5 Poisson Processes and Order Statistics Professor William H. Press, Department of Computer Science,

Probability and Statistics Dr. Saeid Moloudzadeh Poisson Random Variable 1 Contents Descriptive Statistics Axioms of Probability Combinatorial.

Week 61 Poisson Processes Model for times of occurrences (“arrivals”) of rare phenomena where λ – average number of arrivals per time period. X – number.

Random Variable 2013.

Probability Distributions: a review

The Exponential and Gamma Distributions

Expectations of Random Variables, Functions of Random Variables

Probability Theory Overview and Analysis of Randomized Algorithms

ETM 607 – Spreadsheet Simulations

Conditional Probability on a joint discrete distribution

Maximum Likelihood Find the parameters of a model that best fit the data… Forms the foundation of Bayesian inference Slide 1.

Probability & Statistics Probability Theory Mathematical Probability Models Event Relationships Distributions of Random Variables Continuous Random.

Suppose you roll two dice, and let X be sum of the dice. Then X is

Probability Review for Financial Engineers

Distributions and expected value

E370 Statistical analysis for bus & econ

Warm Up Imagine you are rolling 2 six-sided dice. 1) What is the probability to roll a sum of 7? 2) What is the probability to roll a sum of 6 or 7? 3)

Chapter 3 : Random Variables

CSE 321 Discrete Structures

Random Variables A random variable is a rule that assigns exactly one value to each point in a sample space for an experiment. A random variable can be.

1/2555 สมศักดิ์ ศิวดำรงพงศ์

Uniform Probability Distribution

Presentation transcript:

Probability Refresher COMP5416 Advanced Network Technologies

School of Information Technologies COMP5416 Simulation - 2 Discrete random variables Events can take only discrete values, e.g. integer values value Probability Probabilities sum to 1.0

School of Information Technologies COMP5416 Simulation - 3 Example – Roll a die Discrete values 1,2,3,4,5,6 Each with probability 1/6

School of Information Technologies COMP5416 Simulation - 4 Example – Bernoulli random variable Discrete values 1 with probability p, 0 with probability 1- p 01value Probability p 1-p

School of Information Technologies COMP5416 Simulation - 5 Example – Poisson random variable Discrete values 0,1,2,...,infinity Value k has probability p k, with value Probability 07

School of Information Technologies COMP5416 Simulation - 6 Continuous random variables Events can take real values on arbitrary range Probabilities integrate to 1.0

School of Information Technologies COMP5416 Simulation - 7 Continuous random variables Distribution Function

School of Information Technologies COMP5416 Simulation - 8 Example – negative exponential random variable Parameter  Probability density function, negative exponential,  =2 Probability distribution function, negative exponential,  =2

School of Information Technologies COMP5416 Simulation - 9 Example – Gaussian random variable Parameters  Gaussian density function,  =0,  =1 Gaussian distribution function,  =0,  =1