Home  |  Excel  | FrontPage Access Word | PowerPoint  | Outlook | Visio |  Project |  Publisher  |  InfoPath | Exchange  |  SharePoint Commerce  | Silverlight Database  | Data Warehouse  |  SQL  |  OO Design SQL_ServerMYSQL | PHP  | Oracle | VBA ASP.NET  | C programming | Java  |  Photoshop | HTMLDreamweaver | Web DesignExpression Web  | XML  | Java script  |  Security | UML | Flash Silverlight |UNIX | Networking 

Links

www.msoffice.us

dbmanagement.info

italiankitchenrecipes.com

relationshipadvices.info

www.coptics.info

 

  Home  |  Excel  | FrontPage Access Word | PowerPoint  | Outlook | Visio |  Project |  Publisher  |  InfoPath | Exchange  |  SharePoint Commerce  | Silverlight Database  | Data Warehouse  |  SQL  |  OO Design SQL_ServerMYSQL | PHP  | Oracle | VBA ASP.NET  | C programming | Java  |  Photoshop | HTMLDreamweaver | Web DesignExpression Web  | XML  | Java script  |  Security | UML | Flash Silverlight |UNIX | Networking s


MsOffice.US
Home
Excel
FrontPage
Access
Word
PowerPoint
Outlook
Visio
Project
Publisher
InfoPath
Exchange
SharePoint
Commerce
Expression
Silverlight
Management
Software
Windows

 

 

 

 

   

Home Excel Topics
 

An Introduction to Random Variables

 

 


 

 

■     What is a random variable?

 

■     What is a discrete random variable?

 

■     What are the mean, variance, and standard deviation of a random variable?

 

■     What is a continuous random variable?

 

■     What is a probability density function?

 

■     What are independent random variables?

 

In today’s world, the only thing that’s certain is that we face a great deal of uncertainty. In the next nine chapters, I’ll give you some powerful techniques that you can use to incorporate uncertainty in business models. The key building block in modeling uncertainty is under- standing how to use random variables.

 

What is a random variable?

Any situation whose outcome is uncertain is called an experiment. The value of a random variable is based on the (uncertain) outcome of an experiment. For example, tossing a pair of dice is an experiment, and a random variable might be defined as the sum of the values shown on each die. In this case, the random variable could assume any of the val- ues 2, 3, and so on up to 12. As another example, consider the experiment of selling a new video game console, for which a random variable might be defined as the market share for this new product.

 

What is a discrete random variable?

A random variable is discrete if it can assume a finite number of possible values. Here are some examples of discrete random variables:

     Number of potential competitors for your product

 

     Number of aces drawn in a five-card poker hand

 

 

 

 

 

     Number of car accidents you have (hopefully zero!) in a year

 

     Number of dots showing on a die

 

     Number of free throws out of 12 that Steve Nash makes during a basketball game

 

What are the mean, variance, and standard deviation of a random variable?

In Chapter 37, “Summarizing Data with Descriptive Statistics, I discussed the mean, variance, and standard deviation for a data set. In essence, the mean of a random variable

(often denoted by µ) is the average value of the random variable we would expect if we performed an experiment many times. The mean of a random variable is often referred

to as the random variable’s expected value. The variance of a random variable (often denoted by σ2) is the average value of the squared deviation from the mean of a random variable that we would expect if we performed our experiment many times. The stan-

dard deviation of a random variable (often denoted by σ) is simply the square root of its variance. As with data sets, the mean of a random variable is a summary measure for a typical value of the random variable, whereas the variance and standard deviation mea- sure the spread of the random variable about its mean.

 

As an example of how to compute the mean, variance, and standard deviation of a ran- dom variable, suppose we believe that the return on the stock market during the next year is governed by the following probabilities:

 

 

Probability                          Market return

.40                                               +20 percent

.30                                               0 percent

.30                                               -20 percent

 

 

Hand calculations show the following:

 

µ=.40*(.20)+.30*(.00)+.30*(–.20)=.02  or  2  percent

 

σ2=.4*(.20–.02)2+.30*(.0–.02)2+.30*(–.20–.02)2=.0276

 

Then σ=.166 or 16.6 percent.

 

In the file Meanvariance.xlsx (shown in Figure 55-1), I’ve verified these computations.

 

Figure 55-1    Computing the mean, standard deviation, and variance of a random variable


 

 

I computed the mean of our market return in cell C9 with the formula

=SUMPRODUCT(B4:B6,C4:C6). This formula multiplies each value of the random variable by its probability and sums up the products.

 

To compute the variance of our market return, I determined the squared deviation of each value of the random variable from its mean by copying from D4 to D5:D6 the formula =(B4–$C$9)^2. Then, in cell C10, I computed the variance of the market return

as the average squared deviation from the mean with the formula =SUMPRODUCT

(C4:C6,D4:D6). Finally, I computed the standard deviation of the market return in cell

C11 with the formula =SQRT(C10).

 

What is a continuous random variable?

A continuous random variable is a random variable that can assume a very large number or, to all intents and purposes, an infinite number of values. Here are some examples of continuous random variables:

 

     Price of Microsoft stock one year from now

 

     Market share for a new product

 

     Market size for a new product

 

     Cost of developing a new product

 

     Newborn baby’s weight

 

     Person’s IQ

 

     Dirk Nowitzki’s three-point shooting percentage during next season

 

What is a probability density function?

A discrete random variable can be specified by a list of values and the probability of occurrence for each value of the random variable. Because a continuous random variable can assume an infinite number of values, we can’t list the probability of occurrence for each value of a continuous random variable. A continuous random variable is com- pletely described by its probability density function. For example, the probability density function for a randomly chosen person’s IQ is shown in Figure 55-2.

 

Figure 55-2    Probability density function for IQs


 
 

 

 

A probability density function (pdf) has the following properties:

 

     The value of the pdf is always greater than or equal to 0.

 

     The area under the pdf equals 1.

 

     The height of the density function for a value x of a random variable is proportional

to the likelihood that the random variable assumes a value near x. For example, the height of the density for an IQ of 83 is roughly half the height of the density for an IQ

of 100. This tells us that IQs near 83 are approximately half as likely as IQs around

100. Also, because the density peaks at 100, IQs around 100 are most likely.

 

     The  probability  that  a  continuous  random  variable  assumes  a  range  of  values equals the corresponding area under the density function. For example, the frac- tion of people having IQs from 80 through 100 is simply the area under the density from 80 through 100.

What are independent random variables?

A set of random variables are independent if knowledge of the value of any of their sub- sets tells you nothing about the values of the other random variables. For example, the number of games won by the Indiana University football team during a year is indepen- dent of the percentage return on Microsoft during the same year. Knowing that Indiana did very well would not change your view of how Microsoft stock did during the year.

On the other hand, the return on Microsoft stock and Intel stock are not independent.

If we are told that Microsoft stock had a high return in one year, in all likelihood, com- puter sales were high, which tells us that Intel probably had a good year as well.

 

 

Problems

 

1.    Identify the following random variables as discrete or continuous:

 

     Number of games Kerry Wood wins for the Chicago Cubs next season

 

     Number that comes up when spinning a roulette wheel

 

     Unit sales of Tablet PCs next year

 

     Length of time that a light bulb lasts before it burns out

 

2.    Compute the mean, variance, and standard deviation of the number of dots showing when a die is tossed.

3.    Determine whether the following random variables are independent:

 

     Daily temperature and sales at an ice cream store

 

     Suit and number of a card drawn from a deck of playing cards

 

     Inflation and return on the stock market

 

     Price charged for and the number of units sold of a car


 

 

4.    The current price of a company’s stock is $20. The company is a takeover target. If the takeover is successful, the company’s stock price will increase to $30. If the takeover is unsuccessful, the stock price will drop to $12. Determine the range of values for the probability of a successful takeover that would make it worthwhile to purchase the stock today. Assume your goal is to maximize your expected profit. Hint: Use the Microsoft Office Excel 2007 Goal Seek command, which is discussed in detail in Chapter 16, The Goal Seek Command.”

 

 

 


 

List of our Excel topics
(Excel 2003) Advance tutorials
(Excel 2003) Algebra
(Excel 2003) Analyzing external data in Excel
(Excel 2003) Analyzing Your Database
(Excel 2003) Arithmetic
(Excel 2003) Basic Actions
(Excel 2003) Basic Sort Filter
(Excel 2003) Catching data entry errors
(Excel 2003) Charting
(Excel 2003) Charts overview  
(Excel 2003) Conditional Formatting 
(Excel 2003) Constraint and References 
(Excel 2003) Copying Cells
(Excel 2003) Creating a workspace in Excel
(Excel 2003) Creating custom lists
(Excel 2003) Creating Simple Workbook
(Excel 2003) Data Subtotals 
(Excel 2003) DATA VALIDATION  
(Excel 2003) Displaying Symbols
(Excel 2003) Don't use AutoFormat on an entire worksheet
(Excel 2003) EDATE function
(Excel 2003) Ensure proper data entry with validation
(Excel 2003) Establishing Criteria
(Excel 2003) Express yourself with comments
(Excel 2003) Filtering
(Excel 2003) Filtering Database
(Excel 2003) Filters  
(Excel 2003) Format cells  
(Excel 2003) Formatting The Appearance of a Workbook
(Excel 2003) Formatting  Appearance Workbook
(Excel 2003) Functions 
(Excel 2003) Getting Random Sample
(Excel 2003) Guidelines Charting
(Excel 2003) Headers&Footers
(Excel 2003) Hiding duplicate records
(Excel 2003) Hyperlink to a specific cell
(Excel 2003) Insert Date&Time
(Excel 2003) Linear Regression
(Excel 2003) Linking Documents
(Excel 2003) Macro security settings
(Excel 2003) Macros 
(Excel 2003) new features
(Excel 2003) Opening a workbook at startup
(Excel 2003) Page Setup
(Excel 2003) Page set-up Overview 
(Excel 2003) Pivot Tables and Charts
(Excel 2003) PointerShapes
(Excel 2003) Printing
(Excel 2003) Printing overview 
(Excel 2003) Protect a worksheet  
(Excel 2003) Quickly setting a print area
(Excel 2003) Quickly shuffling toolbar buttons 
(Excel 2003) Rank Function
(Excel 2003) SOLVER  
(Excel 2003) Split  Worksheet
(Excel 2003) Statistics
(Excel 2003) Summarize data with grouping
(Excel 2003) Trigonometry
(Excel 2003) text export
(Excel 2003) Using functions for Spreadsheet solutions etc
(Excel 2003) Working PivotTable
(Excel 2003) Working with Clip Art
(Excel 2003) Working with Tables
(Excel 2003) Presentations
(Excel 2003) Step-by-Step Examples
(Excel 2007) Convert Text  Columns
(Excel 2007) Getting Random Sample
(Excel 2007) Getting started Excel
(Excel 2007) New features
(Excel 2007) Running Macros
(Excel 2007) Set Print Titles
(Excel 2007) Split  Worksheet
(Excel 2007) Presentations 
(Excel 2007) An Overview Excel
(Excel 2010) Date-Related Functions
(Excel 2010) Adding Hyperlinks
(Excel 2010) Adding Information Worksheets
(Excel 2010) AutoCorrect AutoFill AutoFit
(Excel 2010) Conditional Formatting
(Excel 2010) Data-entry tips
(Excel 2010) Details the New features
(Excel 2010) Excel (Excel 2010)  Tips Tricks
(Excel 2010) EXCEL SHORT CUTS
(Excel 2010) Formatting Cells
(Excel 2010) Getting Started
(Excel 2010) Grouping Worksheets
(Excel 2010) History Excel
(Excel 2010) IF Statements
(Excel 2010) Inserting Dates Times
(Excel 2010) Internal Rate Return
(Excel 2010) Manipulating Dates Times Text
(Excel 2010) Mastering Art Replacement
(Excel 2010) Moving Worksheets from One Workbook to Another
(Excel 2010) Operator precedence
(Excel 2010) Operator precedence
(Excel 2010) Parts of the Excel Screen
(Excel 2010) Paste Special
(Excel 2010) Smart Formatting Tricks
(Excel 2010) Spell Check
(Excel 2010) Undo Redo AutoRecover
(Excel 2010) What Is Excel Good For
(Excel 2010) What’s new
(Excel 2010) Working with Names
(Excel 2003) Creating Charts
(Excel 2003) Writing Formulas
(Excel 2007) Introduction to charts
(Excel 2003) Using Functions
(Excel 2007) Functions
(Excel 2010) Match Function
(Excel 2010) Text Functions
(Excel 2010) INDEX Function
 Excel Functions Glossary
(Excel 2010) Evaluating Investments
(Excel 2007) Sort filter
(Excel 2010) Range Names
(Excel 2003) Graphing Data Curve Fitting
(Excel 2007) Adding Picture
(Excel 2003) Basic Graphing
(Excel 2010) VBA, Macros, and Other Ways to automate Excel
(Excel 2010) Calculating Various Workbook Elements and Publishing 
(Excel 2010) Consolidating Sheets
(Excel 2010) Function Groups
(Excel 2010) Function Junction
(Excel 2010) Leveraging Excel Functions Using Excel Services
(Excel 2010) Using Excel Functions to Enhance Project Management Productivity
(Excel 2010) The SUMIF, AVERAGEIF, SUMIFS, and AVERAGEIFS Functions
(Excel 2010) The OFFSET Function
(Excel 2010) The Auditing Tool
(Excel 2010) The Goal Seek Command
(Excel 2010) Using the Scenario Manager for Sensitivity Analysis 
(Excel 2010) Financial Functions
(Excel 2010) Circular References
(Excel 2010) COUNTIF, COUNTIFS, COUNT, COUNTA, and COUNTBLANK Functions
(Excel 2010) Sensitivity Analysis with Data Tables
(Excel 2010) An Introduction to Random Variables
(Excel 2010) Analysis of Variance One-Way ANOVA
(Excel 2010) Array Formulas and Functions
(Excel 2010) Calculating an Optimal Bid
(Excel 2010) Consolidating Data
(Excel 2010) Creating Subtotals
(Excel 2010) Determining Customer Value
(Excel 2010) Absolute-relative reference problems
(Excel 2010) Estimating Straight Line Relationships
(Excel 2010) Borrowing and Investing Formulas
(Excel 2010) Forecasting in the Presence of Special Events
(Excel 2010) Examples of Formulas Using the COUNTIF Function
(Excel 2010) Importing Data from a Text File or Document
(Excel 2010) Importing Data from the Internet
(Excel 2010) Incorporating Qualitative Factors into Multiple Regression
(Excel 2010) Introduction to Monte Carlo Simulation
(Excel 2010) Introduction to Multiple Regression
(Excel 2010) Date-Related Functions
(Excel 2010) Modeling Exponential Growth
(Excel 2010) Modeling Nonlinearities and Interactions
(Excel 2010) Nonlinear Pricing
(Excel 2010) Excel Data Types
(Excel 2010) Function Procedure
(Excel 2010) Pricing Stock Options
(Excel 2010) Examples of Formulas Using the COUNTIF Function
(Excel 2010) Randomized Blocks and Two-Way ANOVA
(Excel 2010) Simulating Stock Prices and Asset Allocation Modeling
(Excel 2010) Sorting in Excel
(Excel 2010) Spin Buttons, Scroll Bars, Option Buttons, Check Boxes, Combo Boxes, and Group List Boxes
(Excel 2010) Summarizing Data by Using Descriptive Statistics
(Excel 2010) Summarizing Data by Using Histograms
(Excel 2010) Tables
(Excel 2010) The Binomial, Hypergeometric, and Negative Binomial Random Variables
(Excel 2010) Creating a Custom Number Format
(Excel 2010) The INDIRECT Function
(Excel 2010) The Normal Random Variable
(Excel 2010) The Poisson and Exponential Random Variable
(Excel 2010) The Power Curve
(Excel 2010) Using Correlations to Summarize Relationships
(Excel 2010) Using Moving Averages to Understand Time Series
(Excel 2010) Using PivotTables to Describe Data
(Excel 2010) Creating a Calculated Field or Calculated Item
(Excel 2010) Using Solver for Capital Budgeting
(Excel 2010) Using Solver for Financial Planning
(Excel 2010) Using Solver to Determine the Optimal Product Mix
(Excel 2010) Using Solver to Schedule Your Workforce
(Excel 2010) Using Solver to Solve Transportation or Distribution Problems
(Excel 2010) Validating Data
(Excel 2010) Weibull and Beta istributions: Modeling Machine Life and Duration of a Project
(Excel 2010) Winter’s Method
(Excel 2010) Excel Counting and Summing Functions
(Excel 2010) Formatting dates and times
(Excel 2010) Formula Problems and Solutions
(Excel 2010) Formulas returning an error
(Excel 2010) Miscellaneous Calculations
(Excel 2010) Mismatched parentheses
(Excel 2010) Megaformula Examples
(Excel 2010) Introducing Arrays
(Excel 2010) Pivot table terminology
(Excel 2010) Functions Used in Lookup Formulas
(Excel 2010) Project Management Establishing Project Management Fundamentals
(Excel 2010) Project Management Establishing Excel and Office 2007 SharePoint Server Fundamentals
(Excel 2010) Project Management Initiating the Project
(Excel 2010) Project Management Determining Project Requirements
(Excel 2010) Project Management Planning and Acquiring Resources
(Excel 2010) Project Management Assessing and Tracking Risk
(Excel 2010) Project Management Constructing the Project Schedule and Budget
(Excel 2010) Project Management Establishing Change Control Processes
(Excel 2010) Project Management Controlling Project Outcomes and Archiving Documents
(Excel 2010) Project Management IF Statements.
(Excel 2010) VBA Simulation Basic Tutorial
(Excel 2010) VBA Overview
(Excel 2010) VBA Programming Spreadsheets
(Excel 2010) VBA Programming Language
(Excel 2010) VBA Writing Macros
(Excel 2010) VBA QUE
(Excel 2010) Excel Function Reference
 Excel SOFTWARE SUPPORTING Excel 
(Excel 2010) Several concepts contribute to the time value of money:
(Excel 2010) The Pros and Cons of Megaformulas
(Excel 2010) Types of validation criteria you can apply
(Excel 2010) Using Assignment Expressions
(Excel 2010) Using Lookup Functions
(Excel 2010) Understanding Some Database Terminology
(Excel 2010) The Pros and Cons of Megaformulas
(Excel 2010) Using Built-In VBA Functions
(Excel 2010) Exercises
(Excel 2003) PivotTable_Feature.

 

[Excel/KeywordExcel.htm]

Welcome to

MsOffice.us

This is a free website for learning and practicing all the basics of Microsoft Office 2010 training courses, online help, lessons and more...

Links

www.msoffice.us

dbmanagement.info

italiankitchenrecipes.com

relationshipadvices.info

www.coptics.info

 

 

  Home  Excel  | FrontPage Access Word | PowerPoint  | Outlook | Visio |  Project |  Publisher  |  InfoPath | Exchange  |  SharePoint Commerce  | Silverlight Database  | Data Warehouse  |  SQL  |  OO Design SQL_ServerMYSQL | PHP  | Oracle | VBA ASP.NET  | C programming | Java  |  Photoshop | HTMLDreamweaver | Web DesignExpression Web  | XML  | Java script  |  Security | UML | Flash Silverlight |UNIX | Networking 

  Home  |  Excel  | FrontPage Access Word | PowerPoint  | Outlook | Visio |  Project |  Publisher  |  InfoPath | Exchange  |  SharePoint Commerce  | Silverlight Database  | Data Warehouse  |  SQL  |  OO Design SQL_ServerMYSQL | PHP  | Oracle | VBA ASP.NET  | C programming | Java  |  Photoshop | HTMLDreamweaver | Web DesignExpression Web  | XML  | Java script  |  Security | UML | Flash Silverlight |UNIX | Networking