Discrete Probability Distributions
|
|
|
- Caroline Caldwell
- 6 years ago
- Views:
Transcription
1 Discrete Probability Distributions Answers 1. Suppose a statistician working for CSULA Federal Credit Union collected data on ATM withdrawals for the population of the credit union s customers. The statistician generates the following probability distribution of withdrawals for a specific day. Number of Probability Withdrawals A. Define what the population is. B. Define what the population mean is for the problem. C. Calculate the expected number of ATM withdrawals made on the specific day. D. What is the median number of withdrawals made that day? E. Calculate the variance and standard deviation in the number of withdrawals that day. F. What is P(X>1)? G. What is P(X 2)? 2. A new math book costs $140 and a used one costs $80. A new chemistry book costs $180, a good used one costs $75, and a worn one costs $40. A student wants to buy the cheapest math book and chemistry book available in the bookstore. The probability of getting a used math book is.4, of getting a worn chemistry book is.3, and of getting a good used chemistry book is.2. Let X denote the cost of the two books the student purchases. Assume that the purchases are independent of one another. Construct the probability distribution of X. 3. Suppose you agree to a bet involving a single toss of a fair coin. If the coin winds up heads, you win $200; if it is tails, you lose $100. Define X as the outcome of the bet for you. Construct the probability distribution of X. What is the expected value of X? What is the probability you will lose money? Now suppose the bet involves two coin tosses. The previously stated outcomes apply for each toss (win $200 if heads, lose $100 if tails). Define X as the dollar value of the intersection of events for the coin tosses. Construct the probability distribution of X. What is the expected value of X? What is the probability you will lose money? 4. Farming is a very risky business because of the possibility that bad weather might cause financial ruin. Farmers usually insure their crops each year against possible losses resulting from bad weather. Determine the annual premium for an insurance policy to cover the farmer's $200,000 wheat crop. Insurance representatives believe there is a 3% probability that the crop will be totally destroyed by adverse weather, and a 5% probability of a $100,000 loss due to weather.
2 5. Suppose you have been offered one share of stock from your boss as a Christmas bonus. She offers you a choice of Wal Mart or Sears stock. You know that you are going to hold the stock for one year. Below are the probability distributions of the two stocks prices a year from today. Wal Mart Probability Sears Probability $ $ $50 $ $ $120 A. Fill in the above table. B. What is the expected price for the two stocks? C. What is the variance and standard deviation for the two stocks' prices? D. Why should variance/standard deviation be considered in determining which stock to acquire?
3 Discrete Probability Distributions 1. Suppose a statistician working for CSULA Federal Credit Union collected data on ATM withdrawals for the population of the credit union s customers. The statistician generates the following probability distribution of withdrawals for a specific day. Number of Probability Withdrawals A. Define what the population is. The population is all customers at the CSULA Federal Credit Union. The characteristic we are concerned with is the customers withdrawals on a specific day. B. Define what the population mean is for the problem. The population mean is the average number of withdrawals on a particular for the population. C. Calculate the expected number of ATM withdrawals made on the specific day. The expected number of ATM withdrawals: μ=.74 D. What is the median number of withdrawals made that day? Zero. E. Calculate the variance and standard deviation in the number of withdrawals that day. The population variance is The standard deviation is.934. F. What is P(X>1)?.21 G. What is P(X 2)?.95
4 2. A new math book costs $140 and a used one costs $80. A new chemistry book costs $180, a good used one costs $75, and a worn one costs $40. A student wants to buy the cheapest math book and chemistry book available in the bookstore. The probability of getting a used math book is.4, of getting a worn chemistry book is.3, and of getting a good used chemistry book is.2. Let X denote the cost of the two books the student purchases. Assume that the purchases are independent of one another. Construct the probability distribution of X. X P(X) Suppose you agree to a bet involving a single toss of a fair coin. If the coin winds up heads, you win $200; if it is tails, you lose $100. Define X as the outcome of the bet for you. Construct the probability distribution of X. What is the expected value of X? What is the probability you will lose money? Event X P(X) T H What is the expected value of X? E(X)=$50. What is the probability you will lose money? 0.50 Now suppose the bet involves two coin tosses. The previously stated outcomes apply for each toss (win $200 if heads, lose $100 if tails). Define X as the dollar value of the intersection of events for the coin tosses. Construct the probability distribution of X. Event X P(X) TT TH HT HH What is the expected value of X? E(X)=$100. What is the probability you will lose money? 0.25
5 4. Farming is a very risky business because of the possibility that bad weather might cause financial ruin. Farmers usually insure their crops each year against possible losses resulting from bad weather. Determine the annual premium for an insurance policy to cover the farmer's $200,000 wheat crop. Insurance representatives believe there is a 3% probability that the crop will be totally destroyed by adverse weather, and a 5% probability of a $100,000 loss due to weather. Let X represent the expected loss to the farmer. X P(X) $100, $200, The expected value of X is $11,000. This equals the expected cost to the insurer, in which case the insurance premium will be at least $11, Suppose you have been offered one share of stock from your boss as a Christmas bonus. She offers you a choice of Wal Mart or Sears stock. You know that you are going to hold the stock for one year. Below are the probability distributions of the two stocks prices a year from today. Wal Mart Probability Sears Probability $ $ $50.50 $ $ $ A. Fill in the above table. B. What is the expected price for the two stocks? The expected value of the Wal Mart stock is $60 after a year. The expected value of the Sears stock is $57. C. What is the variance and standard deviation for the two stocks' prices? Wal Mart Sears variance standard deviation
6 D. Why should variance/standard deviation be considered in determining which stock to acquire? The variance and standard deviation are simple measures of the risk of holding a financial asset.
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.
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.
Statistics for Business and Economics: Random Variables (1)
Statistics for Business and Economics: Random Variables (1) STT 315: Section 201 Instructor: Abdhi Sarkar Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides.
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
MATH 118 Class Notes For Chapter 5 By: Maan Omran
MATH 118 Class Notes For Chapter 5 By: Maan Omran Section 5.1 Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Ex1: The test scores
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
Probability: Week 4. Kwonsang Lee. University of Pennsylvania February 13, 2015
Probability: Week 4 Kwonsang Lee University of Pennsylvania kwonlee@wharton.upenn.edu February 13, 2015 Kwonsang Lee STAT111 February 13, 2015 1 / 21 Probability Sample space S: the set of all possible
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
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
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
L04: Homework Answer Key
L04: Homework Answer Key Instructions: You are encouraged to collaborate with other students on the homework, but it is important that you do your own work. Before working with someone else on the assignment,
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
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
Test 7A AP Statistics Name: Directions: Work on these sheets.
Test 7A AP Statistics Name: Directions: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. Suppose X is a random variable with mean µ. Suppose we observe
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
PROBABILITY AND STATISTICS, A16, TEST 1
PROBABILITY AND STATISTICS, A16, TEST 1 Name: Student number (1) (1.5 marks) i) Let A and B be mutually exclusive events with p(a) = 0.7 and p(b) = 0.2. Determine p(a B ) and also p(a B). ii) Let C and
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
Section M Discrete Probability Distribution
Section M Discrete Probability Distribution A random variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted
Probability Distributions. Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution
Probability Distributions Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution Definitions Random Variable: a variable that has a single numerical value
Chapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 2: Mean and Variance of a Discrete Random Variable Section 3.4 1 / 16 Discrete Random Variable - Expected Value In a random experiment,
Bayes s Rule Example. defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?
Bayes s Rule Example A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are
Central Limit Theorem 11/08/2005
Central Limit Theorem 11/08/2005 A More General Central Limit Theorem Theorem. Let X 1, X 2,..., X n,... be a sequence of independent discrete random variables, and let S n = X 1 + X 2 + + X n. For each
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.
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 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
Math 243 Section 4.3 The Binomial Distribution
Math 243 Section 4.3 The Binomial Distribution Overview Notation for the mean, standard deviation and variance The Binomial Model Bernoulli Trials Notation for the mean, standard deviation and variance
Prof. Thistleton MAT 505 Introduction to Probability Lecture 3
Sections from Text and MIT Video Lecture: Sections 2.1 through 2.5 http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systemsanalysis-and-applied-probability-fall-2010/video-lectures/lecture-1-probability-models-and-axioms/
The Mathematics of Normality
MATH 110 Week 9 Chapter 17 Worksheet The Mathematics of Normality NAME Normal (bell-shaped) distributions play an important role in the world of statistics. One reason the normal distribution is important
SECTION 4.4: Expected Value
15 SECTION 4.4: Expected Value This section tells you why most all gambling is a bad idea. And also why carnival or amusement park games are a bad idea. Random Variables Definition: Random Variable A random
X = x p(x) 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6. x = 1 x = 2 x = 3 x = 4 x = 5 x = 6 values for the random variable X
Calculus II MAT 146 Integration Applications: Probability Calculating probabilities for discrete cases typically involves comparing the number of ways a chosen event can occur to the number of ways all
Mean, Variance, and Expectation. Mean
3 Mean, Variance, and Expectation The mean, variance, and standard deviation for a probability distribution are computed differently from the mean, variance, and standard deviation for samples. This section
CHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
CHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS 8.1 Distribution of Random Variables Random Variable Probability Distribution of Random Variables 8.2 Expected Value Mean Mean is the average value of
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
First Name: Last Name: SID: Class Time: M Tu W Th math10 - HW3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Continuous random variables are
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Module 5 Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the specified probability ) Suppose that T is a random variable. Given
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
Random variables. Discrete random variables. Continuous random variables.
Random variables Discrete random variables. Continuous random variables. Discrete random variables. Denote a discrete random variable with X: It is a variable that takes values with some probability. Examples:
Value (x) probability Example A-2: Construct a histogram for population Ψ.
Calculus 111, section 08.x The Central Limit Theorem notes by Tim Pilachowski If you haven t done it yet, go to the Math 111 page and download the handout: Central Limit Theorem supplement. Today s lecture
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
Engineering Statistics ECIV 2305
Engineering Statistics ECIV 2305 Section 5.3 Approximating Distributions with the Normal Distribution Introduction A very useful property of the normal distribution is that it provides good approximations
Example 1: Identify the following random variables as discrete or continuous: a) Weight of a package. b) Number of students in a first-grade classroom
Section 5-1 Probability Distributions I. Random Variables A variable x is a if the value that it assumes, corresponding to the of an experiment, is a or event. A random variable is if it potentially can
LECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE
LECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE MSc Đào Việt Hùng Email: hungdv@tlu.edu.vn Random Variable A random variable is a function that assigns a real number to each outcome in the sample space of a
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
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
Fall 2015 Math 141:505 Exam 3 Form A
Fall 205 Math 4:505 Exam 3 Form A Last Name: First Name: Exam Seat #: UIN: On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work Signature: INSTRUCTIONS Part
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
Lecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
Distribution of the Sample Mean
Distribution of the Sample Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Experiment (1 of 3) Suppose we have the following population : 4 8 1 2 3 4 9 1
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
Business Statistics 41000: Homework # 2
Business Statistics 41000: Homework # 2 Drew Creal Due date: At the beginning of lecture # 5 Remarks: These questions cover Lectures #3 and #4. Question # 1. Discrete Random Variables and Their Distributions
Assignment 2 (Solution) Probability and Statistics
Assignment 2 (Solution) Probability and Statistics Dr. Jitesh J. Thakkar Department of Industrial and Systems Engineering Indian Institute of Technology Kharagpur Instruction Total No. of Questions: 15.
6.1 Discrete & Continuous Random Variables. Nov 4 6:53 PM. Objectives
6.1 Discrete & Continuous Random Variables examples vocab Objectives Today we will... - Compute probabilities using the probability distribution of a discrete random variable. - Calculate and interpret
STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions 5/31/11 Lecture 14 1 Statistic & Its Sampling Distribution
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
Math 180A. Lecture 5 Wednesday April 7 th. Geometric distribution. The geometric distribution function is
Geometric distribution The geometric distribution function is x f ( x) p(1 p) 1 x {1,2,3,...}, 0 p 1 It is the pdf of the random variable X, which equals the smallest positive integer x such that in a
Section Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
Discrete Random Variables
Discrete Random Variables MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives During this lesson we will learn to: distinguish between discrete and continuous
Discrete Random Variables
Discrete Random Variables MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Objectives During this lesson we will learn to: distinguish between discrete and continuous
Chapter 6 Probability
Chapter 6 Probability Learning Objectives 1. Simulate simple experiments and compute empirical probabilities. 2. Compute both theoretical and empirical probabilities. 3. Apply the rules of probability
Expectations. Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or
Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or µ X, is E(X ) = µ X = x D x p(x) Definition Let X be a discrete
Probability Distributions
Chapter 6 Discrete Probability Distributions Section 6-2 Probability Distributions Definitions Let S be the sample space of a probability experiment. A random variable X is a function from the set S into
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.
Section Random Variables and Histograms
Section 3.1 - Random Variables and Histograms Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
Statistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
binomial day 1.notebook December 10, 2013 Probability Quick Review of Probability Distributions!
Probability Binomial Distributions Day 1 Quick Review of Probability Distributions! # boys born in 4 births, x 0 1 2 3 4 Probability, P(x) 0.0625 0.25 0.375 0.25 0.0625 TWO REQUIREMENTS FOR A PROBABILITY
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
Introduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5
Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5 Steve Dunbar Due Fri, October 9, 7. Calculate the m.g.f. of the random variable with uniform distribution on [, ] and then
HUDM4122 Probability and Statistical Inference. February 23, 2015
HUDM4122 Probability and Statistical Inference February 23, 2015 In the last class We studied Bayes Theorem and the Law of Total Probability Any questions or comments? Today Chapter 4.8 in Mendenhall,
Probability Basics. Part 1: What is Probability? INFO-1301, Quantitative Reasoning 1 University of Colorado Boulder. March 1, 2017 Prof.
Probability Basics Part 1: What is Probability? INFO-1301, Quantitative Reasoning 1 University of Colorado Boulder March 1, 2017 Prof. Michael Paul Variables We can describe events like coin flips as variables
Version A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
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
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
FE 5204 Stochastic Differential Equations
Instructor: Jim Zhu e-mail:zhu@wmich.edu http://homepages.wmich.edu/ zhu/ January 13, 2009 Stochastic differential equations deal with continuous random processes. They are idealization of discrete stochastic
Statistics vs. statistics
Statistics vs. statistics Question: What is Statistics (with a capital S)? Definition: Statistics is the science of collecting, organizing, summarizing and interpreting data. Note: There are 2 main ways
HHH HHT HTH THH HTT THT TTH TTT
AP Statistics Name Unit 04 Probability Period Day 05 Notes Discrete & Continuous Random Variables Random Variable: Probability Distribution: Example: A probability model describes the possible outcomes
Theoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
Section Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
MBF2263 Portfolio Management. Lecture 8: Risk and Return in Capital Markets
MBF2263 Portfolio Management Lecture 8: Risk and Return in Capital Markets 1. A First Look at Risk and Return We begin our look at risk and return by illustrating how the risk premium affects investor
d) Find the standard deviation of the random variable X.
Q 1: The number of students using Math lab per day is found in the distribution below. x 6 8 10 12 14 P(x) 0.15 0.3 0.35 0.1 0.1 a) Find the mean for this probability distribution. b) Find the variance
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
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
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
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,
STAT 201 Chapter 6. Distribution
STAT 201 Chapter 6 Distribution 1 Random Variable We know variable Random Variable: a numerical measurement of the outcome of a random phenomena Capital letter refer to the random variable Lower case letters
We use probability distributions to represent the distribution of a discrete random variable.
Now we focus on discrete random variables. We will look at these in general, including calculating the mean and standard deviation. Then we will look more in depth at binomial random variables which are
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.
Chapter 3: Probability Distributions and Statistics
Chapter 3: Probability Distributions and Statistics Section 3.-3.3 3. Random Variables and Histograms A is a rule that assigns precisely one real number to each outcome of an experiment. We usually denote
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
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
Data Analytics (CS40003) Practice Set IV (Topic: Probability and Sampling Distribution)
Data Analytics (CS40003) Practice Set IV (Topic: Probability and Sampling Distribution) I. Concept Questions 1. Give an example of a random variable in the context of Drawing a card from a deck of cards.
MATH20180: Foundations of Financial Mathematics
MATH20180: Foundations of Financial Mathematics Vincent Astier email: vincent.astier@ucd.ie office: room S1.72 (Science South) Lecture 1 Vincent Astier MATH20180 1 / 35 Our goal: the Black-Scholes Formula
Cover Page Homework #8
MODESTO JUNIOR COLLEGE Department of Mathematics MATH 134 Fall 2011 Problem 11.6 Cover Page Homework #8 (a) What does the population distribution describe? (b) What does the sampling distribution of x
1/3/12 AP STATS. WARM UP: How was your New Year? EQ: HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31. Chapter
1/3/12 AP STATS WARM UP: How was your New Year? EQ: Name one way you can tell between discrete variables and continuous variables? HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31 1 Jan. 3, 2012 HW: Pg
Let X be the number that comes up on the next roll of the die.
Chapter 6 - Discrete Probability Distributions 6.1 Random Variables Introduction If we roll a fair die, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6, and each of these numbers has probability
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
5.1 Personal Probability
5. Probability Value Page 1 5.1 Personal Probability Although we think probability is something that is confined to math class, in the form of personal probability it is something we use to make decisions
Chapter 8 Homework Solutions Compiled by Joe Kahlig. speed(x) freq 25 x < x < x < x < x < x < 55 5
H homework problems, C-copyright Joe Kahlig Chapter Solutions, Page Chapter Homework Solutions Compiled by Joe Kahlig. (a) finite discrete (b) infinite discrete (c) continuous (d) finite discrete (e) continuous.
CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS
CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.. The weighted average of the possible
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
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