# The basics of probability theory. Distribution of variables, some important distributions

Save this PDF as:

Size: px
Start display at page:

Download "The basics of probability theory. Distribution of variables, some important distributions"

## Transcription

1 The basics of probability theory. Distribution of variables, some important distributions 1

2 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a coin, rolling a dice, measuring the concentration of a solution, measuring the body weight of an animal, etc. are experiments. Every experiment has more, sometimes infinitely large outcomes 2

3 Event: the result (or outcome) of an experiment elementary events: the possible outcomes of an experiment. composite event: it can be divided into subevents. Example. The experiment is rolling a dice. Elementary events are 1,2,3,4,5,6. Composite events: E1={1,3,5} (the result is an odd number). E2={2,4,6} (the result is an even number). E3={5,6} (the result is greater than 4). Ω={1,2,3,4,5,6} (the result is the certain event). 3

4 Operations with events The complementary event of an event A is the event which occurs whenever A fails. Example: = ={2,4,6} The sum of two events A and B is the event A+B which occurs if A or B occurs. E1+E2={1,3,5}+{2,4,6}={1,2,3,4,5,6} E1+E3={1,3,5}+{5,6}={1,3,5,6} The product of two events A and B is the event AB which occurs if A and B occur. E1 E2={1,3,5} {2,4,6}= E1 E3={1,3,5} {5,6}={5} If A B =, then we say that A and B are mutually exclusive events. 4

5 The concept of probability 5

6 The concept of probability Lets repeat an experiment n times under the same conditions. In a large number of n experiments the event A is observed to occur k times (0 k n). k : frequency of the occurrence of the event A. k/n : relative frequency of the occurrence of the event A. 0 k/n 1 If n is large, k/n will approximate a given number. This number is called the probability of the occurrence of the event A and it is denoted by P(A). 0 P(A) 1 6

7 Example: the tossing of a (fair) coin k: number of head s n= k= k/n= P( head )= k/n 7

8 Probability facts Any probability is a number between 0 and 1. All possible outcomes together must have probability 1. The probability of the complementary event of A is 1-P(A). 8

9 Formula for the calculation of simple exact probabilities 9

10 Rules of probability calculus Assumption: all elementary events are equally probable F P(A) = = T number of favorite outcomes total number of outcomes Examples: Rolling a dice. What is the probability that the dice shows 5? If we let X represent the value of the outcome, then P(X=5)=1/6. What is the probability that the dice shows an odd number? P(odd)=1/2. Here F=3, T=6, so F/T=3/6=1/2. 10

11 Population, sample 11

12 Population, sample Population: the entire group of individuals that we want information about. Sample: a part of the population that we actually examine in order to get information A simple random sample of size n consists of n individuals chosen from the population in such a way that every set of n individuals has an equal chance to be in the sample actually selected. 12

13 Examples Sample data set Questionnaire filled in by a group of pharmacy students Blood pressure of 20 healthy women Population Pharmacy students Students Blood pressure of women (whoever) 13

14 Sample Bar chart of relative frequencies of a categorical variable Population (approximates) Distribution of that variable in the population 100 Gender Percent male female 14

15 Sample Population (approximates) Histogram of relative frequencies of a continuous variable Distribution of that variable in the population 30 Body height 30 Body height Frequency Std. Dev = 8.52 Mean = N = Body height Frequency 10 Std. Dev = 8.52 Mean = N = Body height 15

16 Sample Mean (x) Standard deviation (SD) Median Population (approximates) Mean μ (unknown) Standard deviation σ (unknown) Median (unknown) 30 Body height Frequency Std. Dev = 8.52 Mean = N = Body height 16

17 Random variables, probability distributions (distribution of the population) 17

18 Random variables, probability distributions A random variable is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y,.. Examples The experiment is tossing a coin. X(H)=1 and X(T)=2 Y(H)=-10 and Y(T)=10 The experiment is rolling a dice. Ω={1,2,3,4,5,6}. Let define the X to be the number shown on the dice. 18

19 Distribution of a discrete (categorical) random variable A discrete random variable X has finite number of possible values The probability distribution of X lists the values and their probabilities: Value of X: x 1 x 2 x 3 x n Probability: p 1 p 2 p 3 p n p i 0, p 1 + p 2 +p 3 +p n =1 19

20 Examples The experiment is tossing a coin. p 1 =0.5, p 2 =0.5 The experiment is rolling a dice. p 1 =1/6, p 2 =1/6,, p 6 =1/ x1 x2 0 x1 x2 x3 x4 x5 x6 20

21 Example: rolling two dices Let random variable X be the sum of the two numbers shown on the two dices. P(X=1)=0, (X=1 is impossible) P(X=2)=1/36 (the only favourable event is (1,1), and the number of all possible event is 36. ) j=1 j=2 j=3 j=4 j=5 j=6 i=1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) X i=2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) X i=3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) X i=4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) X i=5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) X i=6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) X / P(X=1) P(X=2) P(X=3) P(X=4) P(X=5) P(X=6) P(X=7) P(X=8) P(X=9) P(X=10) P(X=11) P(X=12) 21

22 Uniform discrete distributions: all p i -s are equal x1 x2 0 x1 x2 x3 x4 x5 x6 Not uniform / P(X=1) P(X=2) P(X=3) P(X=4) P(X=5) P(X=6) P(X=7) P(X=8) P(X=9) P(X=10) P(X=11) P(X=12) 22

23 The binomial distribution Let's consider an experiment A that may have only two possible mutually exclusive outcomes (success, failure) Let P(A)=p We now repeat the experiment n times, let X denote the absolute frequency of the event A. The probability that X will assume any given possible value k is expressed by the binomial formula 23

24 Example Number of successful cases Probability distribution Distribution function Probability of "success" E-06 1 Összesen 1 Probability distribution Distribution function Binomial distribution n=10, input p, k=0,1,,10 In a certain population the occurrence of some disease is p=0.3. What is the probability that examining n=10 patients, there will be exactly k=4 diseased? According to the formula, P(X=4)=10!/(4!6!)(0.3)4(0.7)6= =

25 Continuous random variable A continuous random variable X has takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The density curve is on the above the horizontal axis, and has area exactly 1 underneath it. The probability of any event is the area under the density curve and above the values of X that make up the event. b f ( x) dx = P( a < x < b) a f ( x) dx = 1 P(A) A 25

26 The density curve The density curve is an idealized description of the overall pattern of a distribution that smoothes out the irregularities in the actual data. The density curve is on the above the horizontal axis, and has area exactly 1 underneath it. The area under the curve and above any range of values is the proportion of all observations that fall in that range. P(A) A 26

27 The density curve 1. is on the above the horizontal axis: f(x) 0 The density curve 2. has area exactly 1 underneath it. f ( x) dx =1 a b 3. The area under the curve and above any range of values is the proportion of all observations that fall in that range. b f ( x) dx = P( a x < b) a 27

28 Special density curve: Normal distribution 30 Body height Frequency 0 Std. Dev = 8.52 Mean = N = Body height 28

29 Special distribution of a continuous variable: The normal distribution The normal curve was developed mathematically in 1733 by DeMoivre as an approximation to the binomial distribution. His paper was not discovered until 1924 by Karl Pearson. Laplace used the normal curve in 1783 to describe the distribution of errors. Subsequently, Gauss used the normal curve to analyze astronomical data in The normal curve is often called the Gaussian distribution. The term bell-shaped curve is often used in everyday usage. 29

30 Special distribution of a continuous variable: The normal distribution N(μ,σ 2 ) The density curves are symmetric, single-peaked and bell-shaped The rule. In the normal distribution with mean μ and standard deviation σ: 68% of the observations fall within σ of the mean μ 95% of the observations fall within 2σ of the mean μ 99.7% of the observations fall within 3σ of the mean μ 30

31 Normal distributions N(μ, σ 2 ) N(0,1) N(1,1) 0.6 Probability Density Function y=normal(x;0;1) 0.6 Probability Density Function y=normal(x;1;1) Probability Density Function y=normal(x;0;2) μ, σ : parameters (a parameter is a number that describes the distribution) N(0,2) 31

32 Imagination of the distribution given the sample mean and sample SD supposing normal distribution In the papers generally sample means and SD-s are published. We use these value to know the original distribution For example, age in years: 55.2 ± % of the data are supposed to be in this interval 95.44% of the data are supposed to be in the interval (23.8, 86.6) 32

33 Stent length per lesion (mm): 18.8 ± 10.5 Using these values as parameters, the normal distribution would be like this: Probability Density Function y=normal(x;18.8;10.5) The original distribution of stent length was probably skewed, or if the distribution was normal, there were negative values in the data set. 33

34 Standard normal probabilities x (x): proportion of area to the left of x Probability Density Function y=normal(x;0;1)

35 The central limit theorem 35

36 Distribution of sample means 36

37 The population is not normally distributed 37

38 The central limit theorem If the sample size n is large (say, at least 30), then the population of all possible sample means approximately has a normal distribution with mean µ and standard deviation σ no matter n what probability describes the population sampled 38

39 Tha standard error of mean (SE or SEM) σ is called the standard error of mean n Meaning: the dispersion of the sample means around the (unknown) population mean. 39

40 Calculation of the standard error from the standard deviation when σ is unknown Given x 1, x 2, x 3,, x n statistical sample, the stadard error can be calculated by SE = SD n = n i= 1 ( x i n( n x) 1) 2 It expresses the dispersion of the sample means around the (unknown) population mean. 40

41 Review questions What is the concept of probability? What is the formula for computing simple exact probabilities? When can it be used? What is the distribution of a discrete variable? What are the properties of a discrete distribution? What is the distribution of a continuous variable? What are the properties of the density function? What is the uniform distribution? What is the binomial distribution? What is the normal distribution? Parameters of the normal distribution. Properties of the normal distribution. 41

42 Problems 1. If we roll a dice, there are 6 possible outcomes. If X represents the value of the outcome, find the following probabilities: a) P(X=1);b) P(X>1); c) P(1<X<4) 2. A fair coin is tossed twice. List the possible outcomes? What is the probability of getting two tails? 3. For a standard normal distribution, find the following probabilities using the standard normal table: P(X<0)=. P(X>0)=.. P(X<1)=. P(X>1)=.. P(X<-1)=.. P(-1<X<1)= 42

43 Useful WEB pages g/ats2/normallesson.htm 43

### Histograms and density curves

Histograms and density curves What s in our toolkit so far? Plot the data: histogram (or stemplot) Look for the overall pattern and identify deviations and outliers Numerical summary to briefly describe

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### Chapter 2. The Normal Distribution

Chapter 2 The Normal Distribution Lesson 2-1 Density Curve Review Graph the data Calculate a numerical summary of the data Describe the shape, center, spread and outliers of the data Histogram with Curve

### Continuous Distributions

MAT 2379 3X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take

### Chapter 3 Normal Distribution

Chapter 3 Normal Distribution Density curve A density curve is an idealized histogram, a mathematical model; the curve tells you what values the quantity can take and how likely they are. Example Height

### Probability distributions

Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

### Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

### Discrete and Continuous Random Variables. Summer 2003

Discrete and Continuous Random Variables Summer 003 Random Variables A random variable is a rule that assigns a numerical value to each possible outcome of a probabilistic experiment. We denote a random

### Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### Sampling Distribution of a Normal Variable

Ismor Fischer, 5/9/01 5.-1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,

### Chapter 10: Introducing Probability

Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics

### Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

### Probabilities and Random Variables

Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Lecture 9: Measures of Central Tendency and Sampling Distributions

Lecture 9: Measures of Central Tendency and Sampling Distributions Assist. Prof. Dr. Emel YAVUZ DUMAN Introduction to Probability and Statistics İstanbul Kültür University Faculty of Engineering Outline

### A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

### 4: Probability. What is probability? Random variables (RVs)

4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

### Basics of Probability

Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.

### Math/Stat 370: Engineering Statistics, Washington State University

Math/Stat 370: Engineering Statistics, Washington State University Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 370: Engineering Statistics,

### 3. Continuous Random Variables

3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### University of California, Los Angeles Department of Statistics. Normal distribution

University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Normal distribution The normal distribution is the most important distribution. It describes

### Probability. Gather data by probabilistic (random) mechanism. Use probability to predict results of experiment under assumptions.

Probability Role of probability in statistics: Gather data by probabilistic (random) mechanism. Use probability to predict results of experiment under assumptions. Compute probability of error larger than

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties Theoretical distributions are Theoretical Distributions 1. Binomial distribution 2. Poisson distribution Discrete distribution

### Chapter 6 Random Variables

Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340-344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:

### Monte Carlo Method: Probability

John (ARC/ICAM) Virginia Tech... Math/CS 4414: The Monte Carlo Method: PROBABILITY http://people.sc.fsu.edu/ jburkardt/presentations/ monte carlo probability.pdf... ARC: Advanced Research Computing ICAM:

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### Characteristics of Binomial Distributions

Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

### Lecture 8: Continuous random variables, expectation and variance

Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde

### Comment on the Tree Diagrams Section

Comment on the Tree Diagrams Section The reversal of conditional probabilities when using tree diagrams (calculating P (B A) from P (A B) and P (A B c )) is an example of Bayes formula, named after the

### Probability: Definition and Calculation rules

Statistics for Molecular Medicine -- Probability & Diagnostic Testing -- Barbara Kollerits & Claudia Lamina Medical University of Innsbruck, Division of Genetic Epidemiology Molekulare Medizin, SS 2016

### Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### Probability. Experiment is a process that results in an observation that cannot be determined

Probability Experiment is a process that results in an observation that cannot be determined with certainty in advance of the experiment. Each observation is called an outcome or a sample point which may

### Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

### The Normal Curve. The Normal Curve and The Sampling Distribution

Discrete vs Continuous Data The Normal Curve and The Sampling Distribution We have seen examples of probability distributions for discrete variables X, such as the binomial distribution. We could use it

### Chi-square test Testing for independeny The r x c contingency tables square test

Chi-square test Testing for independeny The r x c contingency tables square test 1 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided

### Probability Models for Continuous Random Variables

Density Probability Models for Continuous Random Variables At right you see a histogram of female length of life. (Births and deaths are recorded to the nearest minute. The data are essentially continuous.)

### 7. Normal Distributions

7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped

### DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS QM 120. Continuous Probability Distribution

DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Dr. Mohammad Zainal Continuous Probability Distribution 2 When a RV x is discrete,

### GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

### Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion

Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

### the number of organisms in the squares of a haemocytometer? the number of goals scored by a football team in a match?

Poisson Random Variables (Rees: 6.8 6.14) Examples: What is the distribution of: the number of organisms in the squares of a haemocytometer? the number of hits on a web site in one hour? the number of

### Lecture 5 : The Poisson Distribution. Jonathan Marchini

Lecture 5 : The Poisson Distribution Jonathan Marchini Random events in time and space Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

### Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination

Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg - ) Chromium (mg kg - ) Copper (mg kg - ) Lead (mg kg - ) Mercury

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### Basic Probability Theory I

A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population

### This HW reviews the normal distribution, confidence intervals and the central limit theorem.

Homework 3 Solution This HW reviews the normal distribution, confidence intervals and the central limit theorem. (1) Suppose that X is a normally distributed random variable where X N(75, 3 2 ) (mean 75

### We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

### REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Math 141. Lecture 5: Expected Value. Albyn Jones 1. jones/courses/ Library 304. Albyn Jones Math 141

Math 141 Lecture 5: Expected Value Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 History The early history of probability theory is intimately related to questions arising

### MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

### Lecture 1: t tests and CLT

Lecture 1: t tests and CLT http://www.stats.ox.ac.uk/ winkel/phs.html Dr Matthias Winkel 1 Outline I. z test for unknown population mean - review II. Limitations of the z test III. t test for unknown population

### STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable

### 6. Distribution and Quantile Functions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 6. Distribution and Quantile Functions As usual, our starting point is a random experiment with probability measure P on an underlying sample spac

### Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1

Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables

### 16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of

### The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### The Chi-Square Distributions

MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

### Sample Term Test 2A. 1. A variable X has a distribution which is described by the density curve shown below:

Sample Term Test 2A 1. A variable X has a distribution which is described by the density curve shown below: What proportion of values of X fall between 1 and 6? (A) 0.550 (B) 0.575 (C) 0.600 (D) 0.625

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### WEEK #22: PDFs and CDFs, Measures of Center and Spread

WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook

### Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

### Exploratory data analysis (Chapter 2) Fall 2011

Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,

### Sample Exam #1 Elementary Statistics

Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### Review. Lecture 3: Probability Distributions. Poisson Distribution. May 8, 2012 GENOME 560, Spring Su In Lee, CSE & GS

Lecture 3: Probability Distributions May 8, 202 GENOME 560, Spring 202 Su In Lee, CSE & GS suinlee@uw.edu Review Random variables Discrete: Probability mass function (pmf) Continuous: Probability density

### Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

### Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.

Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111

### Lecture I. Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions.

Lecture 1 1 Lecture I Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. It is a process consisting of 3 parts. Lecture

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Chapter 1: Looking at Data Distributions. Dr. Nahid Sultana

Chapter 1: Looking at Data Distributions Dr. Nahid Sultana Chapter 1: Looking at Data Distributions 1.1 Displaying Distributions with Graphs 1.2 Describing Distributions with Numbers 1.3 Density Curves

### Variance and Standard Deviation. Variance = ( X X mean ) 2. Symbols. Created 2007 By Michael Worthington Elizabeth City State University

Variance and Standard Deviation Created 2 By Michael Worthington Elizabeth City State University Variance = ( mean ) 2 The mean ( average) is between the largest and the least observations Subtracting

### Math 150 Sample Exam #2

Problem 1. (16 points) TRUE or FALSE. a. 3 die are rolled, there are 1 possible outcomes. b. If two events are complementary, then they are mutually exclusive events. c. If A and B are two independent

### Mathematics Teachers Self Study Guide on the national Curriculum Statement. Book 2 of 2

Mathematics Teachers Self Study Guide on the national Curriculum Statement Book 2 of 2 1 WORKING WITH GROUPED DATA Material written by Meg Dickson and Jackie Scheiber RADMASTE Centre, University of the

### Introduction to Probability

Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

### Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

### Models for Discrete Variables

Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

### Probability and the Binomial Distribution

Probability and the Binomial Distribution Definition: A probability is the chance of some event, E, occurring in a specified manner. NOTATION: P{E} Note: A probability technically is between 0 and 1 (more

### Chapter 6. The Standard Deviation as a Ruler and the Normal Model. Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 6 The Standard Deviation as a Ruler and the Normal Model Copyright 2012, 2008, 2005 Pearson Education, Inc. The Standard Deviation as a Ruler The trick in comparing very different-looking values

### Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

### Let m denote the margin of error. Then

S:105 Statistical Methods and Computing Sample size for confidence intervals with σ known t Intervals Lecture 13 Mar. 6, 009 Kate Cowles 374 SH, 335-077 kcowles@stat.uiowa.edu 1 The margin of error The

### Lecture 4: Probability Spaces

EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 4: Probability Spaces Lecturer: Dr. Krishna Jagannathan Scribe: Jainam Doshi, Arjun Nadh and Ajay M 4.1 Introduction

### MIDTERM EXAMINATION Spring 2009 STA301- Statistics and Probability (Session - 2)

MIDTERM EXAMINATION Spring 2009 STA301- Statistics and Probability (Session - 2) Question No: 1 Median can be found only when: Data is Discrete Data is Attributed Data is continuous Data is continuous