Free Essay

In: Other Topics

Submitted By

Words 789

Pages 4

Words 789

Pages 4

Learning Objectives

1. Understand the concepts of a random variable and a probability distribution.

2. Be able to distinguish between discrete and continuous random variables.

3. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable.

4. Be able to compute and work with probabilities involving a binomial probability distribution.

5. Be able to compute and work with probabilities involving a Poisson probability distribution.

A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals.

Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution n Poisson Distribution

[pic] A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals.

Example: JSL Appliances n Discrete random variable with a finite number of values n Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4)

n Discrete random variable with an infinite sequence of values n Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2, . . . n We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Some Examples:

Example 34:Forty percent of Business travelers carry either cell phone or a laptop. For a sample of 15 business travelers, make the following calculations. a) Compute the probability that three of the travelers carry a cell phone or a laptop. b) Compute the probability that 12 of the travelers carry neither a cell phone nor a laptop c) Compute the probability that at least three of the travelers carry a cell phone or a laptop

Solution a) P(3) = .0634 (from tables)

(b) The answer here is the same as part (a). The probability of 12 failures with p = .60 is the same as the probability of 3 successes with p = .40.

(c). p (3) + p (4) + · · · + p (15) =1 - p (0) - p (1) - p (2) = 1 - .0005 - .0047 - .0219 = .9729

Problem set Chapter 5

1. The manager of a computer repair center in Trenton, New Jersey. has read an advertisement from a local competitor that guarantee. His past service records are used to determine the following probability distribution.

|Number of days |Probability |

|1 |0.15 |

|2 |0.25 |

|3 |0.30 |

|4 |0.18 |

|5 |0.12 |

a. Calculate the mean number of days his customers wait for a computer repair.

b. Also calculate the variance and standard deviation.

c. Based on the calculations in parts a and b, what calculations in parts a and b, what conclusion should the manager reach regarding his company’s repair times?

2. For a binomial distribution with a sample size equal to 10 and a probability of a success equal to 0.30, what is the probability that the sample will contain exactly three successes? Use the calculator to determine the probability.

3. A bank manager has recently examined the credit card account balances for the customers of her bank and found that 20% have an outstanding balance that is at the credit card limit. Suppose the manager randomly selects 15 customers and finds 4 that have balances at the limit. Assume that the properties of the binomial distribution apply. a. What is the probability of finding 4 customers in a sample of 15 who have "maxed out" their credit cards? b. What is the probability that 4 or fewer customers in the sample will have balances at the limit of the credit card?

4. Use the binomial formula to calculate the following probabilities for an experiment in which n = 5 and p = 0.4: a. the probability that x is at most 1 b. the probability that x is at least 4 c. the probability that x is less than 1

5. Studies indicate that 40% of all home buyers will do some remodeling to their home within the first five years of home ownership. If this is true, use the binomial distribution to determine the probability that in a random sample of 20 homeowners, two or fewer will remodel their homes.

-----------------------

.10

.20

.30

.40

0 1 2 3 4

Free Essay

...of Computing and Mathematical Sciences, University of Greenwich September 19, 2012 Dr. Erwin George, Dept. of Mathematics STAT1029 TIPS Invest time learning the language of Mathematics/Statistics/Computing (with all of its special cases and exceptions and conventions). Dr. Erwin George, Dept. of Mathematics STAT1029 TIPS Invest time learning the language of Mathematics/Statistics/Computing (with all of its special cases and exceptions and conventions). Review constantly. Dr. Erwin George, Dept. of Mathematics STAT1029 TIPS Invest time learning the language of Mathematics/Statistics/Computing (with all of its special cases and exceptions and conventions). Review constantly. Do assignments, tutorials, etc. Practise, practise, practise. Dr. Erwin George, Dept. of Mathematics STAT1029 TIPS Invest time learning the language of Mathematics/Statistics/Computing (with all of its special cases and exceptions and conventions). Review constantly. Do assignments, tutorials, etc. Practise, practise, practise. Read lecture notes and try to learn from lectures. Ideally read the lecture material on a topic before the relevant lecture. Dr. Erwin George, Dept. of Mathematics STAT1029 TIPS Invest time learning the language of Mathematics/Statistics/Computing (with all of its special cases and exceptions and conventions). Review constantly. Do assignments, tutorials, etc. Practise, practise, practise. Read lecture notes and try to learn from......

Words: 7020 - Pages: 29

Premium Essay

...PROBABILITY SEDA YILDIRIM 2009421051 DOKUZ EYLUL UNIVERSITY MARITIME BUSINESS ADMINISTRATION CONTENTS Rules of Probability 1 Rule of Multiplication 3 Rule of Addition 3 Classical theory of probability 5 Continuous Probability Distributions 9 Discrete vs. Continuous Variables 11 Binomial Distribution 11 Binomial Probability 12 Poisson Distribution 13 PROBABILITY Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics. There are several competing interpretations of the actual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution. The conditional probability of an event A assuming that B has occurred, denoted ,equals The two faces of probability introduces a central ambiguity which has been around for 350 years and still leads to disagreements about...

Words: 3252 - Pages: 14

Premium Essay

...Introduction to
Statistics
Keone Hon

Words: 11010 - Pages: 45

Free Essay

...ROYAL UNIVERSITY OF PHNOM PENH Master of IT Engineering PROBABILITY AND RANDOM PROCESSES FOR ENGINEERING ASSIGNMENT Topic: BASIC RANDOM PROCESS Group Member: 1, Chor Sophea 2, Lun Sokhemara 3, Phourn Hourheng 4, Chea Daly | Academic year: 2014-2015 I. Introduction Most of the time many systems are best studied using the concept of random variables where the outcome of random experiment was associated with some numerical value. And now there are many more systems are best studied using the concept of multiple random variables where the outcome of a random experiment was associated with multiple numerical values. Here we study random processes where the outcome of a random experiment is associated with a function of time [1]. Random processes are also called stochastic processes. For example, we might study the output of a digital filter being fed by some random signal. In that case, the filter output is described by observing the output waveform at random times. Figure 1.1 The sequence of events leading to assigning a time function x(t) to the outcome of a random experiment Thus a random process assigns a random function of time as the outcome of a random experiment. Figure 1.1 graphically shows the sequence of events leading to assigning a function of time to the outcome of a random experiment.......

Words: 2863 - Pages: 12

Premium Essay

...74537_05_ch05_p215-264.qxd 10/8/12 4:04 PM Page 215 CHAPTER Discrete Probability Distributions CONTENTS STATISTICS IN PRACTICE: CITIBANK 5.1 RANDOM VARIABLES Discrete Random Variables Continuous Random Variables 5.2 DEVELOPING DISCRETE PROBABILITY DISTRIBUTIONS 5.3 EXPECTED VALUE AND VARIANCE Expected Value Variance 5.4 BIVARIATE DISTRIBUTIONS, COVARIANCE, AND FINANCIAL PORTFOLIOS A Bivariate Empirical Discrete Probability Distribution Financial Applications Summary 5.5 BINOMIAL PROBABILITY DISTRIBUTION A Binomial Experiment Martin Clothing Store Problem Using Tables of Binomial Probabilities Expected Value and Variance for the Binomial Distribution POISSON PROBABILITY DISTRIBUTION An Example Involving Time Intervals An Example Involving Length or Distance Intervals HYPERGEOMETRIC PROBABILITY DISTRIBUTION 5 5.6 5.7 74537_05_ch05_p215-264.qxd 10/8/12 4:05 PM Page 219 5.1 Random Variables 219 Exercises Methods SELF test 1. Consider the experiment of tossing a coin twice. a. List the experimental outcomes. b. Define a random variable that represents the number of heads occurring on the two tosses. c. Show what value the random variable would assume for each of the experimental outcomes. d. Is this random variable discrete or continuous? 2. Consider the experiment of a worker assembling a product. a. Define a random variable that represents the time in minutes required to assemble the product. b. What values may the random...

Words: 11596 - Pages: 47

Premium Essay

...outline for MBA (PT) Course name: Statistics Analysis for Business Decisions | Course Code: | Term: Trimester I (Operations Management & Decision Science) | Course Objective:To familiarize the participants on the following: (a) Descriptive Statistics - Concepts / Applications(b) Inferential Statistics – Concepts / Applications(c) Scope and Limitation of Use | Evaluation Criteria: Mid Term Test: 30 marks Test 2 / Quiz: 10 marks Test 3 / Quiz: 10 marks End Term Exam: 50 marks | Faculty: Prof. (Dr.) Tohid Kachwala (email contact tkachwala@nmims.edu, Cabin number: 729, Mobile: 9869166393, Extension: 5871) | Pedagogy: 1. Use of problem solving for all the topics. 2. Use of Statistics in Practice / Case lets / Case studies. 3. Use of Software like Excel / SPSS. | Session Outline: | Session | Topic / Description | 1 | Introduction to Probability – Experiments, Assigning Probabilities, Some basic relationships of Probability Read ASW Chapter 4 or LR Chapter 4 | 2 | Theories of Probability - Classical theory, Relative Frequency theory, Axioms, Addition rule, Multiplication rule, Rule of at least one, Concept of Expected number of Success – Numerical Problems & Applications Case Problem: Hamilton County JudgesSIP: Morton International - Chicago, Illinois Read ASW Chapter 4 or LR Chapter 4 | 3 | Bayes Theorem – Theory, Problems & Applications, Probability revision using tabular......

Words: 845 - Pages: 4

Premium Essay

...Abstract: Statistics represents that body of methods by which characteristics of a population are inferred through observations made in a representative sample from that population. Since scientists rarely observe entire populations, sampling and statistical inference are essential. This article first discusses some general principles for the planning of experiments and data visualization. Then, a strong emphasis is put on the choice of appropriate standard statistical models and methods of statistical inference. (1) Standard models (binomial, Poisson, normal) are described. Application of these models to confidence interval estimation and parametric hypothesis testing are also described, including two-sample situations when the purpose is to compare two (or more) populations with respect to their means or variances. (2) Non-parametric inference tests are also described in cases where the data sample distribution is not compatible with standard parametric distributions. (3) Resampling methods using many randomly computer-generated samples are finally introduced for estimating characteristics of a distribution and for statistical inference. The following section deals with methods for processing multivariate data. Methods for dealing with clinical trials are also briefly reviewed. Finally, a last section discusses statistical computer software and guides the reader through a collection of bibliographic references adapted to different levels of expertise and topics. Statistics......

Words: 4718 - Pages: 19

Premium Essay

...Chapter 6 Statistical Process Control 6.0 Introduction One of the axioms or truisms in law of nature is “No two items of any category at any instant in the universe are the same”. Manufacturing process is no exception to it. It means that variability is part of life and is an inherent property of any process. Measuring, monitoring and managing are rather engineers’ primary job in the global competition. A typical manufacturing scenario can be viewed as shown in the Figure 6.1. That is if one measures the quality characteristic of the output, he will come to know that no two measured characteristics assume same value. This way the variablility conforms one of the axioms or truisms of law of nature; no two items in the universe under any category at any instant will be exactly the same. In maunufacturing scenario, this variability is due to the factors (Random variables) acting upon the input during the process of adding value. Thus the process which is nothing but value adding activity is bound ot experience variability as it is inherent and integral part of the process. Quality had been defined in many ways. Quality is fitness for use is the most common way of looking at it. This fitness for use is governed by the variability. In a maufacturing scenario, despite the fact that a machine operator uses the same precision methods and machines and endeavours to produce identical parts, but the finished products will show a definite variablity. The variability of a......

Words: 3727 - Pages: 15

Premium Essay

...Test1- Answer Key Business Statistics Chapter I: 1.Which scale of measurement can be either numeric or nonnumeric? a. | Nominal | b. | Ratio | c. | Interval | d. | None of these alternatives is correct. | ANS: A PTS: 1 TOP: Descriptive Statistics 2.Which of the following can be classified as quantitative data? a. | interval and ordinal | b. | ratio and ordinal | c. | nominal and ordinal | d. | interval and ratio | ANS: D PTS: 1 TOP: Descriptive Statistics 3.A population is a. | the same as a sample | b. | the selection of a random sample | c. | the collection of all items of interest in a particular study | d. | always the same size as the sample | ANS: C PTS: 1 TOP: Descriptive Statistics 4.On a street, the houses are numbered from 300 to 450. The house numbers are examples of a. | categorical data | b. | quantitative data | c. | both quantitative and categorical data | d. | neither quantitative nor categorical data | ANS: A PTS: 1 TOP: Descriptive Statistics Chapter II: Exhibit 2-1 ========================================================================== The numbers of hours worked (per week) by 400 statistics students are shown below. Number of hours | Frequency | 0 - 9 | 20 | 10 - 19 | 80 | 20 - 29 | 200 | 30 - 39 | 100 | =========================================================================== 5.Refer to Exhibit 2-1. The class width for this distribution a. | is 9 | b....

Words: 1985 - Pages: 8

Premium Essay

...Course Outline: Probability and Statistics Week Tue-Sat 1 2 3 4 Topics/Sub-Topics Introduction to Statistics What is Statistics, Definition of Statistics, Types of Statistics, Application of Statistics in Real life, Variable and its types, Constant and its types. Definition of Data, Primary & Secondary Data, Frequency and Frequency Distribution, Class Limit & Boundary. Organizing and Graphing Data Organizing and Graphing of Qualitative (Simple Bar Chart, Multiple Bar Chart, Percentage Pie Chart) data. Organizing and Graphing of Quantitative data (Stem and Leaf Plot, Histogram, Frequency Polygon, Ogive) Numerical descriptive Measures Measure of central tendency for Ungrouped and Grouped Data (Mean, Median and Mode), Measure of Dispersion and its Types. Cox-Box Plot Variance and use of Standard Deviation, Co-efficient of Variation. Introduction to moment. Week Tue-Sat 5 6 7 8 9 Quizzes/ Assignments Topics/Sub-Topics Moment about origin and Central Moments for Frequency Distribution, Moment Ratios and its interpretation. Introduction to Probability Counting Principle, Probability and its Approaches, Deterministic and non-deterministic Experiment, Sample Space and Events, Outcome, Permutation and Combination. Types of Events Mutually Exclusive Events, Collectively Exhaustive events, Complementary events, Addition Laws of Probability. MID TERM EXAMINATION Assignment 1 Quiz......

Words: 416 - Pages: 2

Premium Essay

...things? X! Daca trebuie sa alegem din n k si conteaza ordinea. How many possibilities? Pnk =n! / (n-k)! Daca trebuie sa alegem din n k si NU conteaza ordinea. How many possibilities? Cnk = n! / ((n-k)!* k!) Relative Frequency = Frequency of A/ Total Frequency = P(A). While statistics (proportions-population) - past events, Probability (sample) - future events. 1) A U B, A or B P(A or B) = P(A) + P(B) - P(A and B) Mutually exclusive => P(A and B)=0 2) P(B/A) Independent=P(B). Dependent=P(B/A)= P(A and B)/ P(A) 3) Intersection A ∩ B ///Dependent :P(A and B) = P(A) x P(B|A) Independent Events: P(A and B) = P(A) x P(B) 4) A stands for “not A” ; Complement Rule: P(A) + P(not A) = 1; P(A) = 1- P(not A); P(At least one) = 1 – P(none) ……………………………………………………………………………………………………………………………………………………………………… ……………………………………………………………………………. 1)UNIFORM DISTRIBUTION: The area under the uniform distribution: P( Mx1 ) z1= 1 - CI% x% (confidence interval dat) of the observation fall below X 3) BINOMIAL DISTRIBUTION: calculez : p(success); n= total no; x = number of successes in sample- ni se da in intrebare; BINOMIAL FORMULA: p(x)= [n!/ (X! (n-x)!)]*px(1-p)n-x ; P(x=x) =P(1)+ P(2)+………………..+P(n) BINOMIAL...

Words: 795 - Pages: 4

Premium Essay

...1. In basic statistics, the three measures of central tendency are mean, median and mode. 2. In basic statistics, the three measures of dispersion are range, variance, and standard deviation. 3. Degrees of freedom are first determined by determining which type of calculation you will be doing that involves degrees of freedom. Likewise, degrees of freedom are calculated differently depending on the type of test you are performing, which is then determined by the number of samples that you have collected. Secondly, take n-1 for only one sample, where n is the number of observations, which is used for estimating variability. This refers to the range of which the scores in a distribution differ from their mean, or average. Thirdly, Use the formula n1 + n2 for two samples, where there are two means to be estimated. Again, n stands for the number of observations in each sample group, which then leaves you with the formula n1 + n2 - 2 degrees of freedom for estimating variability. And finally, use the formula n-p-1 when there are the number n observations and p+1 parameters that are needed to be estimated. 4. The difference in formulas when calculating a population variance and a sample variance is that both variance use different symbols that mean different things. For instance, population variance uses the symbol µ for population mean and N for the population size and when using sample variance for calculations you use x line for the sample mean and n for the number of......

Words: 529 - Pages: 3

Free Essay

...Great Lakes Institute of Management(PGPM 2011-2012) Mid-Term Exam: Time 60 Minutes Statistical Methods for Decision Making Answer as many questions as possible Faculty: P.K. Viswanathan MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Researchers are concerned that the weight of the average American school child is increasing implying, among other things, that children's clothing should be manufactured and marketed in larger sizes. If X is the weight of school children sampled in a nationwide study, then X is an example of A) a categorical random variable. B) a parameter. C) a discrete random variable. D) a continuous random variable. 2) A 99% confidence interval estimate can be interpreted to mean that A) if all possible samples are taken and confidence interval estimates are developed, 99% of them would include the true population mean somewhere within their interval. B) we have 99% confidence that we have selected a sample whose interval does include the population mean. C) Both of the above. D) None of the above. 3) Tim was planning for a meeting with his boss to discuss a raise in his annual salary. In preparation, he wanted to use the Consumer Price Index to determine the percentage increase in his real (inflation-adjusted) salary over the last three years. Which of the 4 methods of data collection was involved when he used the Consumer Price Index? A) Published sources B) Experimentation C) Observation D)......

Words: 3309 - Pages: 14

Premium Essay

...special reference to business management. In addition it will provide necessary statistical knowledge and wide rage of ways to analyze data, which will improve the students statistical analytical and decision making skills. Session Lecture Outline Learning Objectives 01 Basic Probability and Discrete Probability Distributions Simple Probability To develop an understanding of basic probability concepts To introduce conditional probability To use Bayes’ Theorem to revise probabilities in light of new information To provide an understanding of the basic concepts of discrete probability distributions and their characteristics To develop the concept of mathematical expectation for a discrete random variable To introduce the covariance and illustrate its application in finance To present applications of the binomial distribution in business To present applications of the Poisson distribution in business 02 Counting Techniques 03 Continued 04 Conditional Probability 05 Discrete probability distribution. 06 Mathematical Expectation of Discrete Random Variable 07 Properties of mean and variance of a discrete random Variable. 08 Binomial Distribution. 09 Poisson Distribution. 10. The Normal Distribution and Sampling Distribution Introduction Mathematical Models of Continuous Random Variables To...

Words: 1396 - Pages: 6

Premium Essay

...of topics on the exam. The syllabus indicates the exam weights by topic. Copyright 2013 by the Society of Actuaries and the Casualty Actuarial Society C-09-08 PRINTED IN U.S.A. 1. You are given: (i) Losses follow a loglogistic distribution with cumulative distribution function: bx / θ g F b xg = 1+ bx / θ g γ γ (ii) The sample of losses is: 10 35 80 86 90 120 158 180 200 210 1500 Calculate the estimate of θ by percentile matching, using the 40th and 80th empirically smoothed percentile estimates. (A) (B) (C) (D) (E) Less than 77 At least 77, but less than 87 At least 87, but less than 97 At least 97, but less than 107 At least 107 2. You are given: (i) The number of claims has a Poisson distribution. (ii) (iii) (iv) Claim sizes have a Pareto distribution with parameters θ = 0.5 and α = 6 . The number of claims and claim sizes are independent. The observed pure premium should be within 2% of the expected pure premium 90% of the time. Determine the expected number of claims needed for full credibility. (A) (B) (C) (D) (E) Less than 7,000 At least 7,000, but...

Words: 11908 - Pages: 48