Chapter 16 Random Variables Copyright © 2009 Pearson Education, Inc.

Slides:



Advertisements
Similar presentations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide
Advertisements

Copyright © 2010 Pearson Education, Inc. Slide
Chapter 17 Probability Models
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 16 Random Variables.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 17 Probability Models.
Copyright © 2010 Pearson Education, Inc. Chapter 16 Random Variables.
Probability Distributions
Copyright © 2009 Pearson Education, Inc. Chapter 16 Random Variables.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide
1-1 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 1 Chapter 16 Probability Models.
1-1 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 1 Chapter 15 Random Variables.
Slide 1 Statistics Workshop Tutorial 7 Discrete Random Variables Binomial Distributions.
Probability Models Binomial, Geometric, and Poisson Probability Models.
Probability Models Chapter 17.
McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Discrete Random Variables Chapter 4.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 16 Random Variables.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 16 Random Variables.
Chapter 17 Probability Models math2200. I don’t care about my [free throw shooting] percentages. I keep telling everyone that I make them when they count.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Review and Preview This chapter combines the methods of descriptive statistics presented in.
Chapter 7 Lesson 7.5 Random Variables and Probability Distributions
1 Chapter 16 Random Variables. 2 Expected Value: Center A random variable assumes a value based on the outcome of a random event.  We use a capital letter,
Random Variables Chapter 16.
Probability Models Chapter 17.
Chapter 17: probability models
Copyright © 2009 Pearson Education, Inc. Chapter 17 Probability Models.
McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 5 Discrete Random Variables.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 16 Random Variables.
Slide 16-1 Copyright © 2004 Pearson Education, Inc.
Binomial Random Variables Binomial Probability Distributions.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide
Copyright © 2010 Pearson Education, Inc. Chapter 17 Probability Models.
STA 2023 Module 5 Discrete Random Variables. Rev.F082 Learning Objectives Upon completing this module, you should be able to: 1.Determine the probability.
CHAPTER 17 BINOMIAL AND GEOMETRIC PROBABILITY MODELS Binomial and Geometric Random Variables and Their Probability Distributions.
Probability Models Chapter 17. Bernoulli Trials  The basis for the probability models we will examine in this chapter is the Bernoulli trial.  We have.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 17 Probability Models.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 16 Random Variables.
Bernoulli Trials, Geometric and Binomial Probability models.
Chapter 17 Probability Models.
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 5 Discrete Random Variables.
Copyright © 2010 Pearson Education, Inc. Chapter 16 Random Variables.
Slide 17-1 Copyright © 2004 Pearson Education, Inc.
Copyright © 2010 Pearson Education, Inc. Chapter 17 Probability Models.
AP Statistics Probability Models Chapter 17. Objectives: Binomial Distribution –Conditions –Calculate binomial probabilities –Cumulative distribution.
Copyright © 2010 Pearson Education, Inc. Slide
Special Discrete Distributions. Bernoulli Trials The basis for the probability models we will examine in this chapter is the Bernoulli trial. We have.
Statistics 17 Probability Models. Bernoulli Trials The basis for the probability models we will examine in this chapter is the Bernoulli trial. We have.
Copyright © 2009 Pearson Education, Inc. Chapter 17 Probability Models.
Chapter 15 Random Variables.
Binomial Distributions
AP Statistics Probability Models
Chapter 16 Probability Models
Chapter 17 Probability Models Copyright © 2010 Pearson Education, Inc.
Random Variables/ Probability Models
Chapter 15 Random Variables
Chapter 17 Probability Models
Chapter 16 Random Variables.
CHAPTER 16 BINOMIAL, GEOMETRIC, and POISSON PROBABILITY MODELS
Chapter 16 Random Variables
Chapter 16 Probability Models.
Chapter 17 Probability Models Copyright © 2010 Pearson Education, Inc.
Chapter 16 Random Variables.
Chapter 15 Random Variables.
Chapter 16 Probability Models
Chapter 17 Probability Models.
Chapter 16 Random Variables Copyright © 2009 Pearson Education, Inc.
Chapter 16 Random Variables Copyright © 2009 Pearson Education, Inc.
Chapter 17 Probability Models Copyright © 2009 Pearson Education, Inc.
Chapter 16 Random Variables Copyright © 2010 Pearson Education, Inc.
Presentation transcript:

Chapter 16 Random Variables Copyright © 2009 Pearson Education, Inc.

Expected Value: Center A random variable assumes a value based on the outcome of a random event. We use a capital letter, like X, to denote a random variable. A particular value of a random variable will be denoted with a lower case letter, in this case x.

Expected Value: Center (cont.) There are two types of random variables: Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term

Expected Value: Center (cont.) A probability model for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value.

Expected Value: Center (cont.) The expected value of a (discrete) random variable can be found by summing the products of each possible value and the probability that it occurs: Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with.

Example

Chapter 17 Probability Models Copyright © 2009 Pearson Education, Inc.

Bernoulli Trials The basis for the probability models we will examine in this chapter is the Bernoulli trial. We have Bernoulli trials if: there are two possible outcomes (success and failure). the probability of success, p, is constant. the trials are independent.

The Geometric Model A single Bernoulli trial is usually not all that interesting. A Geometric probability model tells us the probability for a random variable that counts the number of Bernoulli trials until the first success. Geometric models are completely specified by one parameter, p, the probability of success, and are denoted Geom(p).

The Geometric Model (cont.) Geometric probability model for Bernoulli trials: Geom(p) p = probability of success q = 1 – p = probability of failure X = number of trials until the first success occurs P(X = x) = (1 – p)x-1p

Independence One of the important requirements for Bernoulli trials is that the trials be independent. When we don’t have an infinite population, the trials are not independent. But, there is a rule that allows us to pretend we have independent trials: The 10% condition: Bernoulli trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population.

Example The American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area. Ex: What is the probability that the fourth blood donor is the first donor with Type B blood?

Example The American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area. Ex #2: What is the probability that the first Type B blood donor is among the first four people in line?

Example Through 2/24/2011 NC State’s free-throw percentage was 69.6%. In the game with GaTech what was the probability that the first missed free-throw by NC State occurs on the 5th attempt? “Success” = missed free throw Success p = 1 - .696 = .304 p(5) = .6964  .304 = .0713

The Binomial Model A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials. Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p).

The Binomial Model (cont.) In n trials, there are ways to have k successes. Read nCk as “n choose k,” and is called a combination. Note: n! = n x (n – 1) x … x 2 x 1, and n! is read as “n factorial.”

The Binomial Model (cont.) Binomial probability model for Bernoulli trials: Binom(n,p) n = number of trials p = probability of success (1 – p) = probability of failure X = number of successes in n trials

What Can Go Wrong? Be sure you have Bernoulli trials. You need two outcomes per trial, a constant probability of success, and independence. Remember that the 10% Condition provides a reasonable substitute for independence. Don’t confuse Geometric and Binomial models.

What have we learned? Bernoulli trials show up in lots of places. Depending on the random variable of interest, we might be dealing with a Geometric model Binomial model

What have we learned? (cont.) Geometric model When we’re interested in the number of Bernoulli trials until the next success. Binomial model When we’re interested in the number of successes in a certain number of Bernoulli trials.