MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS"

Transcription

1 MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution You will find a useful calculation aid for all probability distributions at this web address: This tutorial is a continuation of outcome 4 tutorial 1 D.J.Dunn 1

2 SAMPLE SPACE AND ACCUMULATIVE FREQUENCY Consider the case when two dice are rolled and the outcome is to guess the resulting score. n = 6 and x = so there are 6 = 6 permutations There are only 11 possible scores. You would only need 11 guesses to be sure of getting it right. If we arrange all the possible results into a table we get the sample space shown as the shaded region. First Die Second Die There are 6 events with equal probability so p = 1/6. The probability of getting any score is hence the product of frequency and probability. In cases like this we should use a frequency table as shown and this is called a probability distribution. Score Frequency f P 1/6 /6 /6 4/6 5/6 6/6 5/6 4/6 /6 /6 1/6 Accumulative f 1/6 /6 6/6 10/6 15/6 1/6 6/6 0/6 /6 5/6 6/6 We can see that the probability of getting any given sum is not the same. The only way to get a sum of is to roll a 1 on both dice, but you can get a sum of 7 in six different ways. Plotting shows this is a linear distribution symmetrically placed around the middle value. The accumulative probability always adds up to 1.0. The probability value can only be between 0 and 1. An event that is certain has a probability of 1 and an event that is impossible has a probability of 0. We can see a rule for the probability of a given sum is P = f p f is the frequency of the event and p is the equal probability of any score this being p =1/n r n = number of possibilities for each event (6) and r the number of events (). Hence in this case P = 1/n r x f = 1/6 x f = f/6 It is important to note that the distribution is not a continuous function but a set of discrete values based on the integers 1,,... n D.J.Dunn

3 If we turned the probability distribution into a bar graph with bars of width = 1, the accumulative frequency would be the area of the graph between 0 and the given score. The probability of getting at least 5 is 10/6 from the table and from the graph it is the pink area this being: (1/6 + /6 + /6 + 4/6) = 10/6 The green area represents the probability of getting a score of 6 or more and is simply found by subtracting the pink area from the total area. The total area is always one (6/6). WORKED EXAMPLE No. 1 When two dice are rolled 1 times, what is the probability of guessing correctly a score of 4? What is the probability of guessing a score of least 4 or less? From the plots or table created previously, the probability of guessing a score of exactly 4 is 4/6. The probability of guessing a score of four or less is 10/6 BINOMIAL DISTRIBUTION The Binomial distribution only applies to events where there are two outcomes, say win and lose or heads and tails (tossing a coin). The Binomial Distribution was covered in outcome 1 and was written as: (1 + x) n = 1 + n C 1 x + n C x + n C x + n C 4 x x n The key part is the Binomial coefficient n C r This may be considered as a way of evaluating how many successes 'r' you are likely to get when you repeat the event 'n' times. Let's revise how to evaluate n C r On the top line we put the first r factors of n and on the bottom line we put r! 4 4 x x 4 4 x e.g. C 4 C 6 x x 1 x 1 And if we evaluate these for all values of r we get a symmetrical distribution. The plot shows the result for n = 4. The mean is always the middle value so the mean r is always n/ D.J.Dunn

4 Let's consider the probability distribution for tossing a coin where the probability of a head or tail is both ½ or 0.5. Consider tossing the coin 4 times (n = 4). The sample space is like this. FLIP 1 FLIP FLIP FLIP 4 HEADS H H H H 4 H H H T H H T H H H T T H T H H H T H T H T T H H T T T 1 T H H H T H H T T H T H T H T T 1 T T H H T T H T 1 T T T H 1 T T T T 0 Note there are 4 = 16 possible results. Now we can build the frequency distribution. Note the P = n C r 16 so we didn't need to construct the sample space. Heads (r) n C r P 1/16 4/16 6/16 4/16 1/16 Acc P 1/16 5/16 11/16 15/16 16/16 Suppose we want to know the chances of getting exactly correct. Using r = we see we have a probability of 6/16 = Suppose we want to know the chances of getting or less guesses correct. Using r = we get a probability of 11/16 = Suppose we want the probability of getting or more guesses correct. This would be found by subtracting the last answer from 1 to give 5/16 = WORKED EXAMPLE No. What is the probability of correctly calling four heads when a coin is tossed ten times? The number of possible permutations is 10 = 104 The probability of calling correctly 4 times is n C r / C4 10 x 9 x 8 x 7 1 P x x x x D.J.Dunn 4

5 UNEQUAL PROBABILITY Without proof, when the probability of an event is not 0.5 the probability of getting r results correct n r nr out of n events is: P Crp (1 p) p is the probability of each event. n n In the case p = 0.5 this reduces to P Cr(0.5) which is the same formula already used. If the number of tosses are large (n is large) the frequency distribution resembles a continuous graph and it is tempting to join the points as shown but we should remember that the values of r are integers (whole numbers) and so we can never have values in between. The plot below is for n = 50. For cases where π ½ the distribution becomes skewed. Consider the following case. A bag contains three balls numbered 1 to. A single ball is drawn from the bag at random and then replaced. If this is repeated times we get the following sample space. Note how the pattern is constructed in three groups of 9 giving 7 permutations. Ball drawn Number of ones Ball drawn Number of ones Ball drawn Number of ones Number of times drawn r 0 1 frequency Probability P 8/7 1/7 6/7 1/7 If we make n = 50 we get a curve with a peak at n/ and if p =/ the peak is at n/. n r nr All results are predicted by the equation Pr C p (1 p) r You might try the animation at this web address to see this in action. D.J.Dunn 5

6 WORKED EXAMPLE No. Verify the four results previous for n =, p = 1/ r = n r nr 0 ()() 0 P Crp (1 p) C(1/) (/) (1/) (/) 1/ 7 ()() r = n r nr 1 ()() 1 P Crp (1 p) C(1/) (/) (1/) (/) /9 or 6/7 () r = 1 n r nr 1 () 1 P Crp (1 p) C 1(1/) (/) (1/) (/) 4/9 or 1/7 (1) n r nr r = 0 P C p (1 p) C (1/) (/) 1(1/) (/) 8/7 r WORKED EXAMPLE No A bag contains balls numbered 1, and. One ball is removed at random and noted and then replaced. This is repeated 5 times. What is the probability of guessing the number correctly three times out 5? 0 p = 1/ n = 5 and r = P n r C p (1 p) r nr 5 C (1/) (/) (5)(4)() (1/) ()() (/) MEAN AND VARIANCE OF THE BINOMIAL DISTRIBUTION fx fx In statistics the variance is defined as σ f f In the terminology used here x becomes r and P is the probability of r correct guesses. P r P r σ P P The following example shows that P =1 so this reduces to σ P r P r The standard deviation is σ = S Without proof - It can be shown that this reduces to σ = np(1 - p) and when p = ½ σ = n/4 The mean of the Binomial distribution when p =1/ is clearly the middle value so r = n/ When π ½ we can see from the graphs that the mean is r = pn. When π ½ the standard deviation is σ = np(1-p) D.J.Dunn 6

7 WORKED EXAMPLE No. 5 A coin is flipped six times. Show that the resulting frequency distribution for correct tosses has a standard deviation of 1.5 by use of both formulae. First by the simple method σ = n/4 = 6/4 = 1.5 σ = 1.5 = 1.5 P r P r Next by the full method σ P P r n C r P = n C r / 6 1/64 6/64 15/64 0/64 15/64 6/64 1/64 P = 1 P r 0 6/64 60/64 180/64 40/64 150/64 6/64 P r = 67/64 P r 0/64 6/64 0/64 60/64 60/64 6/64 6/64 P r = 19/64 σ σ = 1.5 P r P P r P WORKED EXAMPLE No. 6 67/ / In the last example, what is the probability of guessing correctly exactly four times and at least four times? From the table we see the probability of guessing four correct is 15/64 but the probability of guessing at least four is ( )/64 = 57/64. This is the accumulative value. WORKED EXAMPLE No. 7 Samples of a product are tested to a certain standard and it is found that there is a probability of 0. that they fail. What is the probability of selecting 5 failures from a selection of 15? What is the mean and standard deviation for this sample? n r nr ! 5 10 p = 0. n = 15 r = 5 P Crp (1 p) C5(0.) (0.8) (0.) (0.8) !10! Mean = pn = 0. x 15 = σ = (15)(0.)(1-0.) = You will find a useful calculating aid for at the following web address D.J.Dunn 7

8 SELF ASSESSMENT EXERCISE No If a coin is tossed 0 times, what is the probability of getting the call correct 5 times? (0.0148). If a six sided die is tossed 10 times, what is the probability of getting the call right five times? (0.01). A lottery system consists of drawing one numbered ball from a bag containing nine. This is repeated with six separate bags. What is the probability of guessing all the numbers drawn? (1/51441) 4. 0 coins are flipped each with a probability of 0.5 that it will be heads. What is the standard deviation for the frequency distribution? (.6) 5. A machine making electrical resistors has a probability of 0.1 that the values will fall outside the target range. What is the probability of randomly picking 0 from a batch of 100 that will be outside the target? ( ) What are the mean and the standard deviation for this distribution? (10 and ) NORMAL DISTRIBUTION CURVES In statistics, the normal distribution is often used. In terms of probability the equation without explanation is given as: rr e /σ P σ π The normal distribution curve is not used exclusively for events with a win/lose or yes/no result but it does give similar results to the Binomial distribution when n is large. The same mean and standard deviation must be used in the comparison. Even for low values of n the curves are well matched as shown in the plotted examples below with n = 10. The normal distribution is not normally used for win/lose situations unless n is 50 or larger. You can compare the Binomial and normal distribution at this web address The normal distribution is more widely used for cases where the standard deviation and mean are known as a result of many measurements. We then use it to predict the probability of a given value or range of values. D.J.Dunn 8

9 The normal distribution curve can be made into one that fits all eventualities. This is done by changing the mean to zero by subtracting r and making the standard deviation 1 by dividing by σ. r r Instead of plotting r we plot z. As this is a standard graph, the area of the graph can be σ tabulated and used to solve problems. The table given here covers the area from - to the value of z. Because the graph is symmetrical, other areas can be worked out as appropriate. The total area is 1.0 so the total either side of the mean is 0.5. Tables of the Normal Distribution Probability Content from - to z Note red area = 1 green area z D.J.Dunn 9

10 WORKED EXAMPLE No. 8 The maximum number of people that can occupy a lift is set at 8. The total weight of 8 people chosen at random follows a normal distribution with a mean of 550 kg and a standard deviation of 150 kg. What is the probability that the total weight of 8 people exceeds 600kg? r r r 550 σ = 150 z 0. 0 σ 150 Look in the table down the left hand column for z = 0. and across under 0.0. The number in the table for z = 0. is 0.69 The green area to the right is = This is the probability that the weight will exceed 600kg. WORKED EXAMPLE No. 9 The lifetime in hours of a mass produced product is represented by the normal distribution curve with a mean of 1400 and a standard deviation of 00. What is the probability that a component taken at random will have a lifetime between 1400 and 1450 hours? r 1400 σ = 00 First find the probability for 1450 hours r r z 0.17 σ 00 From the table P = Next find the probability for 1400 hours r r z 0 σ 00 From the table P = as expected for the mean. The probability of the component having a lifetime between 1400 and 1450 hours is : = D.J.Dunn 10

11 SELF ASSESSMENT EXERCISE No. 1. The height of adult males is normally distributed with a mean of 1.78 m and a standard deviation of m. What is the probability of a randomly selected man having a height of less than 1.6 m? (0.0089). A grinding machine produces components with a mean diameter of 0 mm. All the components are measured and the actual size logged. The standard deviation over a period of time is 0.05 mm. Assuming the normal distribution represents the actual distribution, what is the probability of a component being between 9.95 mm and 0.05 mm diameter? (0.686) (Note this is the standard figure for the range between σ = -1 and σ = +1). The breaking strengths of 150 spot welds was measured in Newton and grouped into bands of 0 N as shown. Range f Calculate the mean and the standard deviation. (Answers N and 9.04 N) Calculate the probability that a sample taken at random will have strength of less than 00 N based on the normal distribution. (Answer about 4%) Calculate the probability based on the raw data above. (Answer 5.%) D.J.Dunn 11

12 POISSON DISTRIBUTION Proof and derivation is not given at this level of study but students will find the derivation of this formula at the following web address. This is a distribution representing discrete samples (same as the Binomial) but it brings the time element into the equation. The probability distribution is given by: λ r e λ P r! r = number of occurrences λ = average occurrences/time interval You will find another useful aid to calculation at this web address. WORKED EXAMPLE No. 10 A business receives order at an average rate of 1 per minute. What is the probability of getting three orders in one minute? λ r 1 e λ e 1 λ = 1 r = P 0.061or 6% r! ()() WORKED EXAMPLE No.11 An emergency service receives an average of.1 false alarms per day. What is the probability of getting four false alarms in a given day? λ r.1 4 e λ e.1 λ =.1 r = 4 P or 10% r! (4)()() SELF ASSESSMENT EXERCISE No. Solve all the following on the assumption that Poisson's distribution applies. 1. On average the demand for a certain product is four per week. If the stock at the beginning of each week is renewed so that there are always 6 in store, what is the probability of running out of stock in any week? (1.4%). A call centre has a capacity to deal with 5 calls per minute on average. What is the probability of getting 0 calls in any minute period? (4.5%). The average time taken for a worker to assemble a certain product is 45 minutes. There are 10 workers employed to make these assemblies. What is the probability of assembling 10 units in an hour? (8%) D.J.Dunn 1

Discrete probability and the laws of chance

Discrete probability and the laws of chance Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

More information

Chapter 6 Continuous Probability Distributions

Chapter 6 Continuous Probability Distributions Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability

More information

Summary of Probability

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

More information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

More information

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

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

More information

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

The basics of probability theory. Distribution of variables, some important distributions The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

More information

Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1

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

More information

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

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

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

Random Variables and Their Expected Values

Random Variables and Their Expected Values Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution

More information

PROBABILITIES AND PROBABILITY DISTRIBUTIONS

PROBABILITIES AND PROBABILITY DISTRIBUTIONS Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL

More information

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,

More information

Practice Problems > 10 10

Practice Problems > 10 10 Practice Problems. A city s temperature measured at 2:00 noon is modeled as a normal random variable with mean and standard deviation both equal to 0 degrees Celsius. If the temperature is recorded at

More information

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

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

More information

The normal approximation to the binomial

The normal approximation to the binomial The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very

More information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

More information

Probability. Experiment - any happening for which the result is uncertain. Outcome the possible result of the experiment

Probability. Experiment - any happening for which the result is uncertain. Outcome the possible result of the experiment Probability Definitions: Experiment - any happening for which the result is uncertain Outcome the possible result of the experiment Sample space the set of all possible outcomes of the experiment Event

More information

PROBABILITY. Chapter Overview Conditional Probability

PROBABILITY. Chapter Overview Conditional Probability PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the

More information

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

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

More information

MATH 201. Final ANSWERS August 12, 2016

MATH 201. Final ANSWERS August 12, 2016 MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

More information

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful.

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful. PROBLEM SET 1 For the first three answer true or false and explain your answer. A picture is often helpful. 1. Suppose the significance level of a hypothesis test is α=0.05. If the p-value of the test

More information

Characteristics of Binomial Distributions

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

More information

Cork Institute of Technology. CIT Mathematics Examination, Paper 2 Sample Paper A

Cork Institute of Technology. CIT Mathematics Examination, Paper 2 Sample Paper A Cork Institute of Technology CIT Mathematics Examination, 2015 Paper 2 Sample Paper A Answer ALL FIVE questions. Each question is worth 20 marks. Total marks available: 100 marks. The standard Formulae

More information

Normal distribution. ) 2 /2σ. 2π σ

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

More information

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

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

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

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

More information

INTRODUCTION TO PROBABILITY AND STATISTICS

INTRODUCTION TO PROBABILITY AND STATISTICS INTRODUCTION TO PROBABILITY AND STATISTICS Conditional probability and independent events.. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a,,, or 4 on the

More information

Probabilistic Strategies: Solutions

Probabilistic Strategies: Solutions Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

More information

PROBABILITY NOTIONS. Summary. 1. Random experiment

PROBABILITY NOTIONS. Summary. 1. Random experiment PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non

More information

3. Continuous Random Variables

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

More information

1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%).

1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%). eview of Introduction to Probability and Statistics Chris Mack, http://www.lithoguru.com/scientist/statistics/review.html omework #2 Solutions 1. Consider an untested batch of memory chips that have a

More information

5.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes. Learning Style

5.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes. Learning Style The Poisson distribution 5.3 Introduction In this block we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and where

More information

Models for Discrete Variables

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

More information

Lecture 5 : The Poisson Distribution. Jonathan Marchini

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,

More information

CHAPTER 7: THE CENTRAL LIMIT THEOREM

CHAPTER 7: THE CENTRAL LIMIT THEOREM CHAPTER 7: THE CENTRAL LIMIT THEOREM Exercise 1. Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews

More information

Math 166:505 Fall 2013 Exam 2 - Version A

Math 166:505 Fall 2013 Exam 2 - Version A Name Math 166:505 Fall 2013 Exam 2 - Version A On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Signature: Instructions: Part I and II are multiple choice

More information

Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2. Hypothesis testing Power Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

More information

Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

More information

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

More information

Math 202-0 Quizzes Winter 2009

Math 202-0 Quizzes Winter 2009 Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

More information

E3: PROBABILITY AND STATISTICS lecture notes

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............................

More information

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

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

More information

Statistics 100 Binomial and Normal Random Variables

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

More information

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

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

More information

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1 IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability Models, by Sheldon

More information

3. Examples of discrete probability spaces.. Here α ( j)

3. Examples of discrete probability spaces.. Here α ( j) 3. EXAMPLES OF DISCRETE PROBABILITY SPACES 13 3. Examples of discrete probability spaces Example 17. Toss n coins. We saw this before, but assumed that the coins are fair. Now we do not. The sample space

More information

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

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,

More information

MAT 1000. Mathematics in Today's World

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

More information

Statistics 1040 Summer 2009 Exam III NAME. Point score Curved Score

Statistics 1040 Summer 2009 Exam III NAME. Point score Curved Score Statistics 1040 Summer 2009 Exam III NAME Point score Curved Score Each question is worth 10 points. There are 12 questions, so a total of 120 points is possible. No credit will be given unless your answer

More information

Topic 5 Review [81 marks]

Topic 5 Review [81 marks] Topic 5 Review [81 marks] A four-sided die has three blue faces and one red face. The die is rolled. Let B be the event a blue face lands down, and R be the event a red face lands down. 1a. Write down

More information

Math 421: Probability and Statistics I Note Set 2

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

More information

AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

More information

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

More information

The normal approximation to the binomial

The normal approximation to the binomial The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There

More information

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

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

More information

Name: Exam III. April 16, 2015

Name: Exam III. April 16, 2015 Department of Mathematics University of Notre Dame Math 10120 Finite Math Spring 2015 Name: Instructors: Garbett & Migliore Exam III April 16, 2015 This exam is in two parts on 10 pages and contains 15

More information

Expected values, standard errors, Central Limit Theorem. Statistical inference

Expected values, standard errors, Central Limit Theorem. Statistical inference Expected values, standard errors, Central Limit Theorem FPP 16-18 Statistical inference Up to this point we have focused primarily on exploratory statistical analysis We know dive into the realm of statistical

More information

GCSE HIGHER Statistics Key Facts

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

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Exam Name 1) Solve the system of linear equations: 2x + 2y = 1 3x - y = 6 2) Consider the following system of linear inequalities. 5x + y 0 5x + 9y 180 x + y 5 x 0, y 0 1) 2) (a) Graph the feasible set

More information

Chi-Square Tests. In This Chapter BONUS CHAPTER

Chi-Square Tests. In This Chapter BONUS CHAPTER BONUS CHAPTER Chi-Square Tests In the previous chapters, we explored the wonderful world of hypothesis testing as we compared means and proportions of one, two, three, and more populations, making an educated

More information

Math 1324 Review Questions for Test 2 (by Poage) covers sections 8.3, 8.4, 8.5, 9.1, 9.2, 9.3, 9.4

Math 1324 Review Questions for Test 2 (by Poage) covers sections 8.3, 8.4, 8.5, 9.1, 9.2, 9.3, 9.4 c Dr. Patrice Poage, March 1, 20 1 Math 1324 Review Questions for Test 2 (by Poage) covers sections 8.3, 8.4, 8.5, 9.1, 9.2, 9.3, 9.4 1. A basketball player has a 75% chance of making a free throw. What

More information

Grade 7/8 Math Circles Fall 2012 Probability

Grade 7/8 Math Circles Fall 2012 Probability 1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics

More information

Chapter 6 Random Variables

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:

More information

Computing Binomial Probabilities

Computing Binomial Probabilities The Binomial Model The binomial probability distribution is a discrete probability distribution function Useful in many situations where you have numerical variables that are counts or whole numbers Classic

More information

Lab 11. Simulations. The Concept

Lab 11. Simulations. The Concept Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant

More information

Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm.

Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm. Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm. Political Science 15 Lecture 12: Hypothesis Testing Sampling

More information

Probability distributions

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,

More information

7.5: Conditional Probability

7.5: Conditional Probability 7.5: Conditional Probability Example 1: A survey is done of people making purchases at a gas station: buy drink (D) no drink (Dc) Total Buy drink(d) No drink(d c ) Total Buy Gas (G) 20 15 35 No Gas (G

More information

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

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

More information

Math 150 Sample Exam #2

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

More information

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day.

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day. Review Exam 2 This is a sample of problems that would be good practice for the exam. This is by no means a guarantee that the problems on the exam will look identical to those on the exam but it should

More information

Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function

Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function Rom Variables Discrete Rom Variables Chs.,, 4 Rom Variables Probability Mass Functions Expectation: The Mean Variance Special Distributions Hypergeometric Binomial Poisson Joint Distributions Independence

More information

Math 3C Homework 3 Solutions

Math 3C Homework 3 Solutions Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard

More information

Review the following from Chapter 5

Review the following from Chapter 5 Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

More information

The Normal Curve. The Normal Curve and The Sampling Distribution

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

More information

Combinations and Permutations

Combinations and Permutations Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination

More information

Monte Carlo Method: Probability

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:

More information

ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

More information

37.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes

37.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes The Poisson distribution 37.3 Introduction In this block we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and

More information

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56 2. Describing Data We consider 1. Graphical methods 2. Numerical methods 1 / 56 General Use of Graphical and Numerical Methods Graphical methods can be used to visually and qualitatively present data and

More information

Chapter 13. Vehicle Arrival Models : Count Introduction Poisson Distribution

Chapter 13. Vehicle Arrival Models : Count Introduction Poisson Distribution Chapter 13 Vehicle Arrival Models : Count 13.1 Introduction As already noted in the previous chapter that vehicle arrivals can be modelled in two interrelated ways; namely modelling how many vehicle arrive

More information

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

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

More information

7.5 Conditional Probability; Independent Events

7.5 Conditional Probability; Independent Events 7.5 Conditional Probability; Independent Events Conditional Probability Example 1. Suppose there are two boxes, A and B containing some red and blue stones. The following table gives the number of stones

More information

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

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

More information

The Chi-Square Distributions

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

More information

WHERE DOES THE 10% CONDITION COME FROM?

WHERE DOES THE 10% CONDITION COME FROM? 1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

More information

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314 Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

More information

Ch. 12.1: Permutations

Ch. 12.1: Permutations Ch. 12.1: Permutations The Mathematics of Counting The field of mathematics concerned with counting is properly known as combinatorics. Whenever we ask a question such as how many different ways can we

More information

Lecture 2 Binomial and Poisson Probability Distributions

Lecture 2 Binomial and Poisson Probability Distributions Lecture 2 Binomial and Poisson Probability Distributions Binomial Probability Distribution l Consider a situation where there are only two possible outcomes (a Bernoulli trial) H Example: u flipping a

More information

1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700

1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700 Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,

More information

Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.

Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation. Probability and the Chi-Square Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system

More information

Chapter 2 - Graphical Summaries of Data

Chapter 2 - Graphical Summaries of Data Chapter 2 - Graphical Summaries of Data Data recorded in the sequence in which they are collected and before they are processed or ranked are called raw data. Raw data is often difficult to make sense

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

More information

PRACTICE PAPER MATHEMATICS Extended Part Module 1 (Calculus and Statistics) Question-Answer Book

PRACTICE PAPER MATHEMATICS Extended Part Module 1 (Calculus and Statistics) Question-Answer Book PP-DSE MATH EP M1 Please stick the barcode label here. HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION Candidate Number PRACTICE PAPER MATHEMATICS Extended

More information

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is. Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

More information

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the

More information

Section 2.1. Tree Diagrams

Section 2.1. Tree Diagrams Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.

More information

People have thought about, and defined, probability in different ways. important to note the consequences of the definition:

People have thought about, and defined, probability in different ways. important to note the consequences of the definition: PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models

More information