Lecture 4: The binomial distribution
|
|
- Polly McKinney
- 5 years ago
- Views:
Transcription
1 Lecture 4: The binomial distribution 4th of November 2015 Lecture 4: The binomial distribution 4th of November / 26
2 Combination and permutation Recapitulatif) Consider 7 students applying to a college for 3 places: Abi Ben Claire Dave Emma Frank Gail How many ways are there of choosing 3 students from 7 when 1 order is important permutation) i.e. Abi, Dave) Dave, Abi) 2 in not important combination) i.e. Abi, Dave)= Dave, Abi) Lecture 4: The binomial distribution 4th of November / 26
3 Permutations There are 3 places to fill at the college and 7 students applied. There will be permutations: = = 7! 4! = 7 P 3. In general, the number of permutations of r objects from n is n P r = n! n r)!. Lecture 4: The binomial distribution 4th of November / 26
4 Combinations For every combination of 3 students, there will be 3! = 6 permutations. For example, the combination Dave, Claire and Abi gives rise to the permutations ACD ADC CAD CDA DAC DCA If we are not interested in the order, the number of combinations of 3 students among 7 is then 7 P 3 7 C 3 =. 3! In general, the number of combinations of r objects from n is n C r = n P r r! = n! n r)!r! Lecture 4: The binomial distribution 4th of November / 26
5 An example of the Binomial distribution An unfair coin: P Head) = 2/3 and P T ail) = 1/3 Let X = No. of heads observed in 5 coin tosses X can take on any of the values 0, 1, 2, 3, 4, 5 X is a discrete random variable Some values of X will be more likely to occur than others. Each value of X will have a probability of occurring. What are these probabilities? Lecture 4: The binomial distribution 4th of November / 26
6 What is P X = 1)? One possible way of observing Head once is if we observe the pattern HT T T T. The probability of obtaining this pattern is PHTTTT) = Lecture 4: The binomial distribution 4th of November / 26
7 There are 32 possible patterns of Head and Tails we might observe. HHHHH THHHH HTHHH HHTHH HHHTH HHHHT TTHHH THTHH THHTH THHHT HTTHH HTHTH HTHHT HHTTH HHTHT HHHTT TTTHH TTHTH TTHHT THTTH THTHT THHTT HTTTH HTTHT HTHTT HHTTT HTTTT THTTT TTHTT TTTHT TTTTH TTTTT Five of the patterns contain just one Head. The other 5 possible combinations all have the same probability so the probability of obtaining one head in 5 coin tosses is ) PX = 1) = ) Lecture 4: The binomial distribution 4th of November / 26
8 What about P X = 2)? This probability can be written as P X = 2) = No. of patterns Probability of pattern 2 ) = 5 C 2 3 3) = In general, the probability to observe x Head and 5 x Tail) is 2 ) x 1 5 x) P X = x) = 5 C x 3 3) Lecture 4: The binomial distribution 4th of November / 26
9 We can use this formula to tabulate the probabilities of each possible value of X. PX = 0) = 5 C 0 2 3) 0 1 3) PX = 1) = 5 C 1 2 3) 1 1 3) PX = 2) = 5 C 2 2 3) 2 1 3) PX = 3) = 5 C 3 2 3) 3 1 3) PX = 4) = 5 C 4 2 3) 4 1 3) PX = 5) = 5 C 5 2 3) 5 1 3) Lecture 4: The binomial distribution 4th of November / 26
10 Distribution of probabilities across the possible values of X. PX) X This situation is a specific example of a Binomial distribution. Lecture 4: The binomial distribution 4th of November / 26
11 Key components of the binomial distribution In general a Binomial distribution arises when we have the following 4 conditions: - n identical trials, e.g. 5 coin tosses - 2 possible outcomes for each trial success and failure, e.g. Heads or Tails - Trials are independent, e.g. each coin toss doesn t affect the others - P success ) = p is the same for each trial, e.g. PHead) = 2/3 is the same for each trial Lecture 4: The binomial distribution 4th of November / 26
12 The binomial distribution If we have the above 4 conditions then if we let X = No. of successes then the probability of observing x successes out of n trials is given by PX = x) = n C x p x 1 p) n x) x = 0, 1,..., n If the probabilities of X are distributed in this way, we write X Binn, p) n and p are called the parameters of the distribution. We say X follows a binomial distribution with parameters n and p. Lecture 4: The binomial distribution 4th of November / 26
13 Example 1 Suppose X Bin10, 0.4), what is PX = 7)? Here we have: n = 10, p = 0.4, x = 7, PX = 7) = 10 C 7 0.4) ) 10 7) = 120)0.4) 7 0.6) Lecture 4: The binomial distribution 4th of November / 26
14 Example 2 Suppose Y Bin8, 0.15), what is PY < 3)? Here we have: n = 8, p = 0.15, PY < 3) = PY = 0) + PY = 1) + PY = 2) = 8 C ) ) C ) ) C ) ) Note that 1 p = Lecture 4: The binomial distribution 4th of November / 26
15 Example 3 Suppose W Bin50, 0.12), what is PW > 2)? Here we have: n = 50, p = 0.12, PW > 2) = PW = 3) + PW = 4) PW = 50) = 1 PW 2) ) = 1 PW = 0) + PW = 1) + PW = 2) = 1 50 C ) ) C ) ) C ) ) 48) ) Note that 1 p = Lecture 4: The binomial distribution 4th of November / 26
16 Different values of n and p lead to different distributions with different shapes: PX) n=10 p= X PX) n=10 p= X PX) n=10 p= X Lecture 4: The binomial distribution 4th of November / 26
17 Expected mean and expected standard deviation We have seen in the first lecture that the sample mean and standard deviation can be used to summarize the shape of a dataset. In the case of a probability distribution we have no data as such so we must use the probabilities to calculate the expected mean and standard deviation. Lecture 4: The binomial distribution 4th of November / 26
18 Example: X Bin5, 2/3) Consider the example of the Binomial distribution we saw above x PX = x) The expected mean value of the distribution, denoted µ can be calculated as µ = ) ) ) ) = ) ) Lecture 4: The binomial distribution 4th of November / 26
19 Expected mean and expected standard deviation In general, there is a formula for the mean of a Binomial distribution. There is also a formula for the standard deviation, σ. If X Binn, p) then µ = np σ = npq where q = 1 p In the example above, X Bin5, 2/3) and so the mean and standard deviation are given by µ = np = 5 2/3) = and σ = npq = 5 2/3) 1/3) = Lecture 4: The binomial distribution 4th of November / 26
20 Testing a hypotheses using the Binomial distribution An example Consider the following simple situation: You have a six-sided die, and you have the impression that it s somehow been weighted so that the number 1 comes up more frequently than it should. How would you decide whether this impression is correct? Lecture 4: The binomial distribution 4th of November / 26
21 You could do a careful experiment, where you roll the die 60 times, and count how often the 1 comes up. Suppose you do the experiment, and the 1 comes up 30 times and other numbers come up 30 times all together). If the die is unbiased, you expect the 1 to come up one time in six, i.e. 10 times. Therefore 30 times seems high. But is it too high? There are two possible hypotheses: 1 The die is biased. 2 Just by chance we got more 1 s than expected. How do we decide between these possibilities? Lecture 4: The binomial distribution 4th of November / 26
22 Perform an hypothesis test. Hypothesis: The die is fair. All 6 outcomes have the same probability. Experiment: We roll the die 60 times. Sample: We obtain 60 outcomes and the 1 comes out 30 times. Assuming our hypothesis is true the experiment we carried out satisfies the conditions of the Binomial distribution n identical trials, i.e. 60 die rolls. 2 possible outcomes for each trial: 1 and not 1. Trials are independent. P success ) = 1/6 is the same for each trial Lecture 4: The binomial distribution 4th of November / 26
23 We define X = No. of 1 s that come up. We observed X = 30. We can calculate the probability of observing X=30 if our hypothesis is true, i.e. if X Bin60,1/6): P X = 30) = 60 C ) 30 ) Conclusion: Under the hypothesis that the die is fair, the probability that the number of 1 s come up 30 times in this experiment is very low. Therefore we may conclude that the die has been biased. Lecture 4: The binomial distribution 4th of November / 26
24 Hypothesis testing Now we summarise the general approach: posit a hypothesis design and carry out an experiment to collect a sample of data test to see if the sample is consistent with the hypothesis Testing the hypothesis: Assuming our hypothesis is true what is the probability that we would have observed such a sample or a sample more extreme, i.e. is our sample quite unlikely to have occurred under the assumptions of our hypothesis? Lecture 4: The binomial distribution 4th of November / 26
25 Example: Drug efficiency Until recently an average of 60 out 100 patients have survived a particular severe infection. When a new drug was administered to 15 patients with the infection, 12 of them survived. Does this provide evidence that the new drug is effective? Lecture 4: The binomial distribution 4th of November / 26
26 Hypothesis: The drug is not effective, i.e. the probability of surviving is still p = 0.6. Experiment: We test the drug on 15 patients with the infection. Sample: 12 patients survived. Let X denote the number of patients who survived. Under our hypothesis, X Bin15,0.6) We compute the probability that we would have observed such a sample assuming our hypothesis is true: P X = 12) = 15 C ) ) There is more than 6% chance of observing such a number of surviving patients if the drug in not effective. Therefore it may be just by chance that we observe such a number of patients who survived. Lecture 4: The binomial distribution 4th of November / 26
Lecture 4 : The Binomial Distribution
objects from Permutations and Combinations (Recap) Lecture 4 : The Binomial Distribution Jonathan Marchini Consider 7 students applying to a college for 3 places Abi Ben Claire Dave Emma Frank Gail How
TOPIC P6: PROBABILITY DISTRIBUTIONS SPOTLIGHT: THE HAT CHECK PROBLEM
TOPIC P6: PROBABILITY DISTRIBUTIONS SPOTLIGHT: THE HAT CHECK PROBLEM Some time ago, it was common for men to wear hats when they went out for dinner. When one entered a restaurant, each man would give
Fixed number of n trials Independence
The Binomial Setting Binomial Distributions IB Math SL - Santowski Fixed number of n trials Independence Two possible outcomes: success or failure Same probability of a success for each observation If
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
MA 1125 Lecture 12 - Mean and Standard Deviation for the Binomial Distribution. Objectives: Mean and standard deviation for the binomial distribution.
MA 5 Lecture - Mean and Standard Deviation for the Binomial Distribution Friday, September 9, 07 Objectives: Mean and standard deviation for the binomial distribution.. Mean and Standard Deviation of the
(# of die rolls that satisfy the criteria) (# of possible die rolls)
BMI 713: Computational Statistics for Biomedical Sciences Assignment 2 1 Random variables and distributions 1. Assume that a die is fair, i.e. if the die is rolled once, the probability of getting each
Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances
Physical Principles in Biology Biology 3550 Fall 2018 Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances Monday, 10 September 2018 c David P. Goldenberg University
chapter 13: Binomial Distribution Exercises (binomial)13.6, 13.12, 13.22, 13.43
chapter 13: Binomial Distribution ch13-links binom-tossing-4-coins binom-coin-example ch13 image Exercises (binomial)13.6, 13.12, 13.22, 13.43 CHAPTER 13: Binomial Distributions The Basic Practice of Statistics
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
Section 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
Binomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
The normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
Lecture 8 - Sampling Distributions and the CLT
Lecture 8 - Sampling Distributions and the CLT Statistics 102 Kenneth K. Lopiano September 18, 2013 1 Basics Improvements 2 Variability of Estimates Activity Sampling distributions - via simulation Sampling
MA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
The Binomial distribution
The Binomial distribution Examples and Definition Binomial Model (an experiment ) 1 A series of n independent trials is conducted. 2 Each trial results in a binary outcome (one is labeled success the other
Random Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES
Random Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES Essential Question How can I determine whether the conditions for using binomial random variables are met? Binomial Settings When the
The Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
Chapter 6: Random Variables
Chapter 6: Random Variables Section 6.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 Transforming and
Statistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
Midterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
No, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
Simple Random Sample
Simple Random Sample A simple random sample (SRS) of size n consists of n elements from the population chosen in such a way that every set of n elements has an equal chance to be the sample actually selected.
MATH1215: Mathematical Thinking Sec. 08 Spring Worksheet 9: Solution. x P(x)
N. Name: MATH: Mathematical Thinking Sec. 08 Spring 0 Worksheet 9: Solution Problem Compute the expected value of this probability distribution: x 3 8 0 3 P(x) 0. 0.0 0.3 0. Clearly, a value is missing
A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
5.4 Normal Approximation of the Binomial Distribution
5.4 Normal Approximation of the Binomial Distribution Bernoulli Trials have 3 properties: 1. Only two outcomes - PASS or FAIL 2. n identical trials Review from yesterday. 3. Trials are independent - probability
Examples: Random Variables. Discrete and Continuous Random Variables. Probability Distributions
Random Variables Examples: Random variable a variable (typically represented by x) that takes a numerical value by chance. Number of boys in a randomly selected family with three children. Possible values:
Random Variables. Chapter 6: Random Variables 2/2/2014. Discrete and Continuous Random Variables. Transforming and Combining Random Variables
Chapter 6: Random Variables Section 6.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Random Variables 6.1 6.2 6.3 Discrete and Continuous Random Variables Transforming and Combining
Chapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
Module 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
Chapter 6: Random Variables. Ch. 6-3: Binomial and Geometric Random Variables
Chapter : Random Variables Ch. -3: Binomial and Geometric Random Variables X 0 2 3 4 5 7 8 9 0 0 P(X) 3???????? 4 4 When the same chance process is repeated several times, we are often interested in whether
MA 1125 Lecture 18 - Normal Approximations to Binomial Distributions. Objectives: Compute probabilities for a binomial as a normal distribution.
MA 25 Lecture 8 - Normal Approximations to Binomial Distributions Friday, October 3, 207 Objectives: Compute probabilities for a binomial as a normal distribution.. Normal Approximations to the Binomial
Binomal and Geometric Distributions
Binomal and Geometric Distributions Sections 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 7-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu
Probability Distributions for Discrete RV
Probability Distributions for Discrete RV Probability Distributions for Discrete RV Definition The probability distribution or probability mass function (pmf) of a discrete rv is defined for every number
Statistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
Math 14 Lecture Notes Ch. 4.3
4.3 The Binomial Distribution Example 1: The former Sacramento King's DeMarcus Cousins makes 77% of his free throws. If he shoots 3 times, what is the probability that he will make exactly 0, 1, 2, or
Shifting our focus. We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why?
Probability Introduction Shifting our focus We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why? What is Probability? Probability is used
Probability & Statistics Chapter 5: Binomial Distribution
Probability & Statistics Chapter 5: Binomial Distribution Notes and Examples Binomial Distribution When a variable can be viewed as having only two outcomes, call them success and failure, it may be considered
5.4 Normal Approximation of the Binomial Distribution Lesson MDM4U Jensen
5.4 Normal Approximation of the Binomial Distribution Lesson MDM4U Jensen Review From Yesterday Bernoulli Trials have 3 properties: 1. 2. 3. Binomial Probability Distribution In a binomial experiment with
Chapter 8: Binomial and Geometric Distributions
Chapter 8: Binomial and Geometric Distributions Section 8.1 Binomial Distributions The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Section 8.1 Binomial Distribution Learning Objectives
Part 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
ECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
Stat511 Additional Materials
Binomial Random Variable Stat511 Additional Materials The first discrete RV that we will discuss is the binomial random variable. The binomial random variable is a result of observing the outcomes from
Elementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
Chapter 5 Student Lecture Notes 5-1. Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 5 Student Lecture Notes 5-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals
Central Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
Stat 211 Week Five. The Binomial Distribution
Stat 211 Week Five The Binomial Distribution Last Week E x E x = x p(x) = n p σ x = x μ x 2 p(x) We will see this again soon!! Binomial Experiment We have an experiment with the following qualities : 1.
Lecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4 - Would the correlation between x and y in the table above be positive or negative? The correlation is negative. -
Random Variables and Probability Functions
University of Central Arkansas Random Variables and Probability Functions Directory Table of Contents. Begin Article. Stephen R. Addison Copyright c 001 saddison@mailaps.org Last Revision Date: February
ECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic Probability Distributions: Binomial and Poisson Distributions Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College
CHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Binomial and Geometric Random
Sampling. Marc H. Mehlman University of New Haven. Marc Mehlman (University of New Haven) Sampling 1 / 20.
Sampling Marc H. Mehlman marcmehlman@yahoo.com University of New Haven (University of New Haven) Sampling 1 / 20 Table of Contents 1 Sampling Distributions 2 Central Limit Theorem 3 Binomial Distribution
Binomial and multinomial distribution
1-Binomial distribution Binomial and multinomial distribution The binomial probability refers to the probability that a binomial experiment results in exactly "x" successes. The probability of an event
MATH 264 Problem Homework I
MATH Problem Homework I Due to December 9, 00@:0 PROBLEMS & SOLUTIONS. A student answers a multiple-choice examination question that offers four possible answers. Suppose that the probability that the
Binomial Random Variables
Models for Counts Solutions COR1-GB.1305 Statistics and Data Analysis Binomial Random Variables 1. A certain coin has a 25% of landing heads, and a 75% chance of landing tails. (a) If you flip the coin
Binomial Distributions
Binomial Distributions Binomial Experiment The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials There are only two possible outcomes of interest
Chapter 9. Sampling Distributions. A sampling distribution is created by, as the name suggests, sampling.
Chapter 9 Sampling Distributions 9.1 Sampling Distributions A sampling distribution is created by, as the name suggests, sampling. The method we will employ on the rules of probability and the laws of
AP Statistics Ch 8 The Binomial and Geometric Distributions
Ch 8.1 The Binomial Distributions The Binomial Setting A situation where these four conditions are satisfied is called a binomial setting. 1. Each observation falls into one of just two categories, which
The Central Limit Theorem. Sec. 8.2: The Random Variable. it s Distribution. it s Distribution
The Central Limit Theorem Sec. 8.1: The Random Variable it s Distribution Sec. 8.2: The Random Variable it s Distribution X p and and How Should You Think of a Random Variable? Imagine a bag with numbers
Math 14 Lecture Notes Ch The Normal Approximation to the Binomial Distribution. P (X ) = nc X p X q n X =
6.4 The Normal Approximation to the Binomial Distribution Recall from section 6.4 that g A binomial experiment is a experiment that satisfies the following four requirements: 1. Each trial can have only
Lecture 7 Random Variables
Lecture 7 Random Variables Definition: A random variable is a variable whose value is a numerical outcome of a random phenomenon, so its values are determined by chance. We shall use letters such as X
Probability is the tool used for anticipating what the distribution of data should look like under a given model.
AP Statistics NAME: Exam Review: Strand 3: Anticipating Patterns Date: Block: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%-30%) Probability is the tool used
Chapter 4 Discrete Random variables
Chapter 4 Discrete Random variables A is a variable that assumes numerical values associated with the random outcomes of an experiment, where only one numerical value is assigned to each sample point.
guessing Bluman, Chapter 5 2
Bluman, Chapter 5 1 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question.
Stat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 13: Binomial Formula Tessa L. Childers-Day UC Berkeley 14 July 2014 By the end of this lecture... You will be able to: Calculate the ways an event can
STA 6166 Fall 2007 Web-based Course. Notes 10: Probability Models
STA 6166 Fall 2007 Web-based Course 1 Notes 10: Probability Models We first saw the normal model as a useful model for the distribution of some quantitative variables. We ve also seen that if we make a
ME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
STA258H5. Al Nosedal and Alison Weir. Winter Al Nosedal and Alison Weir STA258H5 Winter / 41
STA258H5 Al Nosedal and Alison Weir Winter 2017 Al Nosedal and Alison Weir STA258H5 Winter 2017 1 / 41 NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION. Al Nosedal and Alison Weir STA258H5 Winter 2017
Math 361. Day 8 Binomial Random Variables pages 27 and 28 Inv Do you have ESP? Inv. 1.3 Tim or Bob?
Math 361 Day 8 Binomial Random Variables pages 27 and 28 Inv. 1.2 - Do you have ESP? Inv. 1.3 Tim or Bob? Inv. 1.1: Friend or Foe Review Is a particular study result consistent with the null model? Learning
Discrete Probability Distribution
1 Discrete Probability Distribution Key Definitions Discrete Random Variable: Has a countable number of values. This means that each data point is distinct and separate. Continuous Random Variable: Has
Binomial distribution
Binomial distribution Jon Michael Gran Department of Biostatistics, UiO MF9130 Introductory course in statistics Tuesday 24.05.2010 1 / 28 Overview Binomial distribution (Aalen chapter 4, Kirkwood and
Lecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability
Chapter 4 Discrete Random variables
Chapter 4 Discrete Random variables A is a variable that assumes numerical values associated with the random outcomes of an experiment, where only one numerical value is assigned to each sample point.
Introduction to Probability and Inference HSSP Summer 2017, Instructor: Alexandra Ding July 19, 2017
Introduction to Probability and Inference HSSP Summer 2017, Instructor: Alexandra Ding July 19, 2017 Please fill out the attendance sheet! Suggestions Box: Feedback and suggestions are important to the
4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
CHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Binomial and Geometric Random
5.2 Random Variables, Probability Histograms and Probability Distributions
Chapter 5 5.2 Random Variables, Probability Histograms and Probability Distributions A random variable (r.v.) can be either continuous or discrete. It takes on the possible values of an experiment. It
Binomial and Geometric Distributions
Binomial and Geometric Distributions Section 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH Department of Mathematics University of Houston February 11, 2016
FINAL REVIEW W/ANSWERS
FINAL REVIEW W/ANSWERS ( 03/15/08 - Sharon Coates) Concepts to review before answering the questions: A population consists of the entire group of people or objects of interest to an investigator, while
TOPIC: PROBABILITY DISTRIBUTIONS
TOPIC: PROBABILITY DISTRIBUTIONS There are two types of random variables: A Discrete random variable can take on only specified, distinct values. A Continuous random variable can take on any value within
A random variable is a quantitative variable that represents a certain
Section 6.1 Discrete Random Variables Example: Probability Distribution, Spin the Spinners Sum of Numbers on Spinners Theoretical Probability 2 0.04 3 0.08 4 0.12 5 0.16 6 0.20 7 0.16 8 0.12 9 0.08 10
MA 1125 Lecture 14 - Expected Values. Wednesday, October 4, Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Wednesday, October 4, 27 Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
Part 10: The Binomial Distribution
Part 10: The Binomial Distribution The binomial distribution is an important example of a probability distribution for a discrete random variable. It has wide ranging applications. One readily available
The binomial distribution p314
The binomial distribution p314 Example: A biased coin (P(H) = p = 0.6) ) is tossed 5 times. Let X be the number of H s. Fine P(X = 2). This X is a binomial r. v. The binomial setting p314 1. There are
Statistics Class 15 3/21/2012
Statistics Class 15 3/21/2012 Quiz 1. Cans of regular Pepsi are labeled to indicate that they contain 12 oz. Data Set 17 in Appendix B lists measured amounts for a sample of Pepsi cans. The same statistics
Normal Approximation to Binomial Distributions
Normal Approximation to Binomial Distributions Charlie Vollmer Department of Statistics Colorado State University Fort Collins, CO charlesv@rams.colostate.edu May 19, 2017 Abstract This document is a supplement
Chapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
Week 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
Business Statistics. Chapter 5 Discrete Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
Chapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 9. Introduction This chapter connects the material in Chapters 4 through 8 (numerical descriptive statistics, sampling, and probability distributions, in particular) with
Econ 6900: Statistical Problems. Instructor: Yogesh Uppal
Econ 6900: Statistical Problems Instructor: Yogesh Uppal Email: yuppal@ysu.edu Lecture Slides 4 Random Variables Probability Distributions Discrete Distributions Discrete Uniform Probability Distribution
BINOMIAL EXPERIMENT SUPPLEMENT
BINOMIAL EXPERIMENT SUPPLEMENT Binomial Experiment - 1 A binomial experiment is any situation that involves n trials with each trial having one of two possible outcomes (Success or Failure) and the probability
Lecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
Assignment 3 - Statistics. n n! (n r)!r! n = 1,2,3,...
Assignment 3 - Statistics Name: Permutation: Combination: n n! P r = (n r)! n n! C r = (n r)!r! n = 1,2,3,... n = 1,2,3,... The Fundamental Counting Principle: If two indepndent events A and B can happen
S = 1,2,3, 4,5,6 occurs
Chapter 5 Discrete Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. Discrete random variables associated with these experiments
4.2 Bernoulli Trials and Binomial Distributions
Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan 4.2 Bernoulli Trials and Binomial Distributions A Bernoulli trial 1 is an experiment with exactly two outcomes: Success and
Using the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the
Using the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the mean, use the CLT for the mean. If you are being asked to
work to get full credit.
Chapter 18 Review Name Date Period Write complete answers, using complete sentences where necessary.show your work to get full credit. MULTIPLE CHOICE. Choose the one alternative that best completes the
Chapter 8. Binomial and Geometric Distributions
Chapter 8 Binomial and Geometric Distributions Lesson 8-1, Part 1 Binomial Distribution What is a Binomial Distribution? Specific type of discrete probability distribution The outcomes belong to two categories
Lecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution