text:
## Sorting correlation coefficients by their magnitudes in a SAS macro

#### Theoretical Background

Many statisticians and data scientists use the correlation coefficient to study the relationship between 2 variables.  For 2 random variables, $X$ and $Y$, the correlation coefficient between them is defined as their covariance scaled by the product of their standard deviations.  Algebraically, this can be expressed as

$\rho_{X, Y} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} = \frac{E[(X - \mu_X)(Y - \mu_Y)]}{\sigma_X \sigma_Y}$.

In real life, you can never know what the true correlation coefficient is, but you can estimate it from data.  The most common estimator for $\rho$ is the Pearson correlation coefficient, which is defined as the sample covariance between $X$ and $Y$ divided by the product of their sample standard deviations.  Since there is a common factor of

$\frac{1}{n - 1}$

in the numerator and the denominator, they cancel out each other, so the formula simplifies to

$r_P = \frac{\sum_{i = 1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i - \bar{x})^2 \sum_{i = 1}^{n}(y_i - \bar{y})^2}}$.

In predictive modelling, you may want to find the covariates that are most correlated with the response variable before building a regression model.  You can do this by

1. computing the correlation coefficients
2. obtaining their absolute values
3. sorting them by their absolute values.

## Potato Chips and ANOVA, Part 2: Using Analysis of Variance to Improve Sample Preparation in Analytical Chemistry

In this second article of a 2-part series on the official JMP blog, I use analysis of variance (ANOVA) to assess a sample-preparation scheme for quantifying sodium in potato chips.  I illustrate the use of the “Fit Y by X” platform in JMP to implement ANOVA, and I propose an alternative sample-preparation scheme to obtain a sample with a smaller variance.  This article is entitled “Potato Chips and ANOVA, Part 2: Using Analysis of Variance to Improve Sample Preparation in Analytical Chemistry“.

If you haven’t read my first blog post in this series on preparing the data in JMP and using the “Stack Columns” function to transpose data from wide format to long format, check it out!  I presented this topic at the last Vancouver SAS User Group (VanSUG) meeting on Wednesday, November 4, 2015.

My thanks to Arati Mejdal, Louis Valente, and Mark Bailey at JMP for their guidance in writing this 2-part series!  It is a pleasure to be a guest blogger for JMP!

## Potato Chips and ANOVA in Analytical Chemistry – Part 1: Formatting Data in JMP

I am very excited to write again for the official JMP blog as a guest blogger!  Today, the first article of a 2-part series has been published, and it is called “Potato Chips and ANOVA in Analytical Chemistry – Part 1: Formatting Data in JMP“.  This series of blog posts will talk about analysis of variance (ANOVA), sampling, and analytical chemistry, and it uses the quantification of sodium in potato chips as an example to illustrate these concepts.

The first part of this series discusses how to import the data into the JMP and prepare them for ANOVA.  Specifically, it illustrates how the “Stack Columns” function is used to transpose the data from wide format to long format.

I will present this at the Vancouver SAS User Group (VanSUG) meeting later today.

## Odds and Probability: Commonly Misused Terms in Statistics – An Illustrative Example in Baseball

Yesterday, all 15 home teams in Major League Baseball won on the same day – the first such occurrence in history.  CTV News published an article written by Mike Fitzpatrick from The Associated Press that reported on this event.  The article states, “Viewing every game as a 50-50 proposition independent of all others, STATS figured the odds of a home sweep on a night with a full major league schedule was 1 in 32,768.”  (Emphases added)

Screenshot captured at 5:35 pm Vancouver time on Wednesday, August 12, 2015.

Out of curiosity, I wanted to reproduce this result.  This event is an intersection of 15 independent Bernoulli random variables, all with the probability of the home team winning being 0.5.

$P[(\text{Winner}_1 = \text{Home Team}_1) \cap (\text{Winner}_2 = \text{Home Team}_2) \cap \ldots \cap (\text{Winner}_{15}= \text{Home Team}_{15})]$

Since all 15 games are assumed to be mutually independent, the probability of all 15 home teams winning is just

$P(\text{All 15 Home Teams Win}) = \prod_{n = 1}^{15} P(\text{Winner}_i = \text{Home Team}_i)$

$P(\text{All 15 Home Teams Win}) = 0.5^{15} = 0.00003051757$

Now, let’s connect this probability to odds.

It is important to note that

• odds is only applicable to Bernoulli random variables (i.e. binary events)
• odds is the ratio of the probability of success to the probability of failure

For our example,

$\text{Odds}(\text{All 15 Home Teams Win}) = P(\text{All 15 Home Teams Win}) \ \div \ P(\text{At least 1 Home Team Loses})$

$\text{Odds}(\text{All 15 Home Teams Win}) = 0.00003051757 \div (1 - 0.00003051757)$

$\text{Odds}(\text{All 15 Home Teams Win}) = 0.0000305185$

The above article states that the odds is 1 in 32,768.  The fraction 1/32768 is equal to 0.00003051757, which is NOT the odds as I just calculated.  Instead, 0.00003051757 is the probability of all 15 home teams winning.  Thus, the article incorrectly states 0.00003051757 as the odds rather than the probability.

This is an example of a common confusion between probability and odds that the media and the general public often make.  Probability and odds are two different concepts and are calculated differently, and my calculations above illustrate their differences.  Thus, exercise caution when reading statements about probability and odds, and make sure that the communicator of such statements knows exactly how they are calculated and which one is more applicable.

## How to Extract a String Between 2 Characters in R and SAS

#### Introduction

I recently needed to work with date values that look like this:

 mydate Jan 23/2 Aug 5/20 Dec 17/2

I wanted to extract the day, and the obvious strategy is to extract the text between the space and the slash.  I needed to think about how to program this carefully in both R and SAS, because

1. the length of the day could be 1 or 2 characters long
2. I needed a code that adapted to this varying length from observation to observation
3. there is no function in either language that is suited exactly for this purpose.

In this tutorial, I will show you how to do this in both R and SAS.  I will write a function in R and a macro program in SAS to do so, and you can use the function and the macro program as you please!

## Using PROC SQL to Find Uncommon Observations Between 2 Data Sets in SAS

A common task in data analysis is to compare 2 data sets and determine the uncommon rows between them.  By “uncommon rows”, I mean rows whose identifier value exists in one data set but not the other. In this tutorial, I will demonstrate how to do so using PROC SQL.

Let’s create 2 data sets.

data dataset1;
input id $group$ gender $age; cards; 111 A Male 11 111 B Male 11 222 D Male 12 333 E Female 13 666 G Female 14 999 A Male 15 999 B Male 15 999 C Male 15 ; run; data dataset2; input id$ group $gender$ age;
cards;
111 A Male 11
999 C Male 15
;
run;

First, let’s identify the observations in dataset1 whose ID variable values don’t exist in dataset2.  I will export this set of observations into a data set called mismatches1, and I will print it for your viewing.  The logic of the code is simple – find the IDs in dataset1 that are not in the IDs in dataset2.  The code “select *” ensures that all columns from dataset1 are used to create the data set in mismatches1.

## Separating Unique and Duplicate Observations Using PROC SORT in SAS 9.3 and Newer Versions

As Fareeza Khurshed commented in my previous blog post, there is a new option in SAS 9.3 and later versions that allows sorting and the identification of duplicates to be done in one step.  My previous trick uses FIRST.variable and LAST.variable to separate the unique observations from the duplicate observations, but that requires sorting the data set first before using the DATA step to do the separation.  If you have SAS 9.3 or a newer version, here is an example of doing it in one step using PROC SORT.

There is a data set called ADOMSG in the SASHELP library that is built into SAS.  It has an identifier called MSGID, and there are duplicates by MSGID.  Let’s create 2 data sets out of SASHELP.ADOMSG:

• DUPLICATES for storing the duplicate observations
• SINGLES for storing the unique observations
proc sort
out = duplicates
uniqueout = singles
nouniquekey;
by msgid;
run;

Here is the log:

NOTE: There were 459 observations read from the data set SASHELP.ADOMSG.
NOTE: 300 observations with unique key values were deleted.
NOTE: The data set WORK.DUPLICATES has 159 observations and 6 variables.
NOTE: The data set WORK.SINGLES has 300 observations and 6 variables.
NOTE: PROCEDURE SORT used (Total process time):
real time 0.28 seconds
cpu time 0.00 seconds

Note that the number of observations in WORK.DUPLICATES and WORK.SINGLES add to 459, the total number of observations in the original data set.

In addition to Fareeza, I also thank CB for sharing this tip.

## Resources for Learning Data Manipulation in R, SAS and Microsoft Excel

I had the great pleasure of speaking to the Department of Statistics and Actuarial Science at Simon Fraser University on last Friday to share my career advice with its students and professors.  I emphasized the importance of learning skills in data manipulation during my presentation, and I want to supplement my presentation by posting some useful resources for this skill.  If you are new to data manipulation, these are good guides for how to get started in R, SAS and Microsoft Excel.

For R, I recommend Winston Chang’s excellent web site, “Cookbook for R“.  It has a specific section on manipulating data; this is a comprehensive list of the basic skills that every data analyst and statistician should learn.

For SAS, I recommend the UCLA statistical computing web page that is adapted from Oliver Schabenberger’s web site.

For Excel, I recommend Excel Easy, a web site that was started at the University of Amsterdam in 2010.  It is a good resource for learning about Excel in general, and there is no background required.  I specifically recommend the “Functions” and “Data Analysis” sections.

A blog called teachr has a good list of Top 10 skills in Excel to learn.

I like to document tips and tricks for R and SAS that I like to use often, especially if I struggled to find them on the Internet.  I encourage you to check them out from time to time, especially in my “Data Analysis” category.

If you have any other favourite resources for learning data manipulation or data analysis, please share them in the comments!

## The advantages of using count() to get N-way frequency tables as data frames in R

#### Introduction

I recently introduced how to use the count() function in the “plyr” package in R to produce 1-way frequency tables in R.  Several commenters provided alternative ways of doing so, and they are all appreciated.  Today, I want to extend that tutorial by demonstrating how count() can be used to produce N-way frequency tables in the list format – this will magnify the superiority of this function over other functions like table() and xtabs().

#### 2-Way Frequencies: The Cross-Tabulated Format vs. The List-Format

To get a 2-way frequency table (i.e. a frequency table of the counts of a data set as divided by 2 categorical variables), you can display it in a cross-tabulated format or in a list format.

In R, the xtabs() function is good for cross-tabulation.  Let’s use the “mtcars” data set again; recall that it is a built-in data set in Base R.

> y = xtabs(~ cyl + gear, mtcars)
> y
gear
cyl      3     4     5
4        1     8     2
6        2     4     1
8        12    0     2

## How to Get the Frequency Table of a Categorical Variable as a Data Frame in R

#### Introduction

One feature that I like about R is the ability to access and manipulate the outputs of many functions.  For example, you can extract the kernel density estimates from density() and scale them to ensure that the resulting density integrates to 1 over its support set.

I recently needed to get a frequency table of a categorical variable in R, and I wanted the output as a data table that I can access and manipulate.  This is a fairly simple and common task in statistics and data analysis, so I thought that there must be a function in Base R that can easily generate this.  Sadly, I could not find such a function.  In this post, I will explain why the seemingly obvious table() function does not work, and I will demonstrate how the count() function in the ‘plyr’ package can achieve this goal.

#### The Example Data Set – mtcars

Let’s use the mtcars data set that is built into R as an example.  The categorical variable that I want to explore is “gear” – this denotes the number of forward gears in the car – so let’s view the first 6 observations of just the car model and the gear.  We can use the subset() function to restrict the data set to show just the row names and “gear”.

> head(subset(mtcars, select = 'gear'))
gear
Mazda RX4            4
Mazda RX4 Wag        4
Datsun 710           4
Hornet 4 Drive       3
Valiant              3

## Exploratory Data Analysis – All Blog Posts on The Chemical Statistician

This series of posts introduced various methods of exploratory data analysis, providing theoretical backgrounds and practical examples.  Fully commented and readily usable R scripts are available for all topics for you to copy and paste for your own analysis!  Most of these posts involve data visualization and plotting, and I include a lot of detail and comments on how to invoke specific plotting commands in R in these examples.

Useful R Functions for Exploring a Data Frame

The 5-Number Summary – Two Different Methods in R

Combining Histograms and Density Plots to Examine the Distribution of the Ozone Pollution Data from New York in R

Conceptual Foundations of Histograms – Illustrated with New York’s Ozone Pollution Data

Quantile-Quantile Plots for New York’s Ozone Pollution Data

Kernel Density Estimation and Rug Plots in R on Ozone Data in New York and Ozonopolis

2 Ways of Plotting Empirical Cumulative Distribution Functions in R

Conceptual Foundations of Empirical Cumulative Distribution Functions

Combining Box Plots and Kernel Density Plots into Violin Plots for Ozone Pollution Data

Kernel Density Estimation – Conceptual Foundations

Variations of Box Plots in R for Ozone Concentrations in New York City and Ozonopolis

Computing Descriptive Statistics in R for Data on Ozone Pollution in New York City

How to Get the Frequency Table of a Categorical Variable as a Data Frame in R

The advantages of using count() to get N-way frequency tables as data frames in R

## Calculating the sum or mean of a numeric (continuous) variable by a group (categorical) variable in SAS

#### Introduction

A common task in data analysis and statistics is to calculate the sum or mean of a continuous variable.  If that variable can be categorized into 2 or more classes, you may want to get the sum or mean for each class.

This sounds like a simple task, yet I took a surprisingly long time to learn how to do this in SAS and get exactly what I want – a new data with with each category as the identifier and the calculated sum/mean as the value of a second variable.  Here is an example to show you how to do it using PROC MEANS.

Read more to see an example data set and get the SAS code to calculate the sum or mean of a continuous variable by a categorical variable!


text:
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# Journal of Display Technology

## Filter Results

Displaying Results 1 - 24 of 24
• ### Front Cover

Publication Year: 2016, Page(s): C1
| PDF (886 KB)
• ### Journal of Display Technology publication information

Publication Year: 2016, Page(s): C2
| PDF (66 KB)

Publication Year: 2016, Page(s):665 - 666
| PDF (60 KB)
• ### Four-Domain Electrode Structure for Wide Viewing Angle in a Fringe-Field-Switching Liquid Crystal Display

Publication Year: 2016, Page(s):667 - 672
| | PDF (1053 KB) | HTML

Although the fringe-field-switching (FFS) mode based on a two-domain (2D) electrode structure has been widely used for wide viewing angle in a liquid crystal display (LCD), it still suffers from color shift and asymmetric luminance distribution at off-axis. We propose a four-domain (4D) electrode structure for wide viewing angle characteristics in an FFS LCD. We demonstrate that the use of the 4D ... View full abstract»

• ### Content-Dependent Reduction of Static Color Breakup on Field Sequential Color LCDs

Publication Year: 2016, Page(s):673 - 680
| | PDF (2223 KB) | HTML

Field sequential color (FSC) LCDs form color images by sequentially displaying subfield images. Because color filter attenuating light transmission is not required, FSC LCDs can reduce power consumption considerably. However, their sequential display scheme results in undesirable color artifacts that degrade the perceived image quality. This paper proposes a new content-dependent static color brea... View full abstract»

• ### Frequency Selection for High Efficiency Wireless Power Transfer

Publication Year: 2016, Page(s):681 - 684
Cited by:  Papers (1)
| | PDF (730 KB)

The analysis of wireless power transfer efficiency is currently limited to a priori choice of spot frequencies. This paper presents a wireless power transfer system for applications in displays which examine transfer efficiency for a broad range of resonant frequencies. The system provides a dc voltage output, achieved using self-oscillation in coupled inductor-capacitor resonator pair. Measuremen... View full abstract»

• ### Effects of Illumination on the Noise Behavior of Amorphous Indium Gallium Zinc Oxide Thin Film Transistors

Publication Year: 2016, Page(s):685 - 689
Cited by:  Papers (2)
| | PDF (1146 KB) | HTML

In this paper, we investigate the response of low-frequency noise to the light for the amorphous indium gallium zinc oxide (a-IGZO) thin film transistors (TFTs). Compared with the case of nonillumination, the noise level increases with the light intensity. We find that the induced noise is more related to the drain current ( ID) with or without illumination, even though the corresponding gate volt... View full abstract»

• ### Numerical Simulation and Analysis of the Switching Performance for Printable Low-Voltage Organic Thin-Film Transistors in Active-Matrix Backplanes

Publication Year: 2016, Page(s):690 - 694
| | PDF (739 KB) | HTML

To achieve low-voltage organic thin-film transistors (OTFTs) with printing or coating processes over large area, a thick gate dielectric layer is preferred. Taking this into consideration, a common solution for low-voltage OTFTs is using large-permittivity (high- k) dielectric material to enlarge the gate dielectric capacitance. Our recent work shows that reducing the subgap density of states (DOS... View full abstract»

• ### Computation and Display of 3D Movie From a Single Integral Photography

Publication Year: 2016, Page(s):695 - 700
Cited by:  Papers (2)
| | PDF (1217 KB) | HTML Media

Integral photography is an auto-stereoscopic technique that allows, among other interesting applications, the display of 3D images with full parallax and avoids the painful effects of the accommodation-convergence conflict. Currently, one of the main drawbacks of this technology is the need of a huge amount of data, which have to be stored and transmitted. This is due to the fact that behind every... View full abstract»

• ### Restoring Integral Images from Focal Stacks Using Compressed Sensing Techniques

Publication Year: 2016, Page(s):701 - 706
| | PDF (2742 KB) | HTML Media

This paper contains an original development of the compressed sensing technique for restoring integral images from a number of observed 2D images. The proposed data acquisition uses a conventional camera equipped with a horizontal 1D mask placed in the pupil plane of the lens. The compressed sensing style algorithm developed is based on a sparsity hypothesis imposed on 2D cross sections of the lig... View full abstract»

• ### Simulation and Control of Display Uniformity in a Backlight Illuminated Image Array

Publication Year: 2016, Page(s):707 - 714
Cited by:  Papers (1)
| | PDF (1511 KB) | HTML

The relative retinal illumination (RRI) distribution of a display system consisting of arrayed backlight units, focusing lenses, and viewing screen is simulated with Monte Carlo ray tracing method. A random scattering model is applied to describe the optical seam effect in the vicinity of the stitching. RRI distribution on the screen viewed at different locations for an autostereoscopic display sy... View full abstract»

• ### Autostereoscopic Eccentric Projection Display with Adjustable Image Sizes and Viewing Zones

Publication Year: 2016, Page(s):715 - 720
| | PDF (1216 KB) | HTML

In this paper, a new autostereoscopic projection display using eccentric Fresnel projection lenses is proposed, which can integrate all parallax image sources on a single panel and is simpler than conventional autostereoscopic projection systems. The use of eccentric Fresnel projection lenses also guarantees intrinsically jointed exit pupils, which allow viewers to adjust the viewing zones based o... View full abstract»

• ### Scanning Prism with Polymer-Stabilized Blue Phase Liquid Crystals

Publication Year: 2016, Page(s):721 - 726
| | PDF (1111 KB) | HTML

A scanning prism with polymer-stabilized blue phase liquid crystals (PS-BPLCs) was demonstrated with an impulsive blinking backlight system of high-resolution three-dimensional (3-D) display. The fast switching PS-BPLC prism, whose response time was less than 1 ms, may solve the resolution degradation problem for the autostereoscopic display. Also, the impulsive blinking backlight system may impro... View full abstract»

• ### A Novel Optimization Method for Lenticular 3-D Display Based on Light Field Decomposion

Publication Year: 2016, Page(s):727 - 735
Cited by:  Papers (7)
| | PDF (1687 KB) | HTML

Crosstalk is a primary defect in affecting the image quality of stereoscopic three-dimensional (3-D) displays. Until now, the crosstalk reduction methods either require extra devices or need tedious calibration procedures, which require precise measurement on each display device. We propose herein a new method of synthesizing lenticular 3-D display based on the light field decomposition and optimi... View full abstract»

• ### Study of Low-Efficiency Droop in Semipolar ( $20\bar{2}\bar{1}$) InGaN Light-Emitting Diodes by Time-Resolved Photoluminescence

Publication Year: 2016, Page(s):736 - 741
Cited by:  Papers (9)
| | PDF (906 KB) | HTML

The superior low-efficiency droop performance of semipolar ($20\bar{2}\bar{1}$) InGaN light-emitting diodes (LEDs) makes it a hot candidate for efficient solid-state lighting and full-color displays. To unveil the mystery of this low droop and high efficiency, the emission dynamics of semipolar ( View full abstract»

• ### Full-Color Pixelated-Addressable Light Emitting Diode on Transparent Substrate (LEDoTS) Micro-Displays by CoB

Publication Year: 2016, Page(s):742 - 746
Cited by:  Papers (4)
| | PDF (951 KB) | HTML

In this paper, we report the design and fabrication of full-color pixelated-addressable light emitting diode on transparent substrate (LEDoTS) micro-displays. The LEDoTS micro-displays were realized by integrating monochromatic LEDs on a transparent quartz substrate using chip on board (CoB) technique. The full-color pixel has a pitch of 1 mm (P1) and includes red, green, and blue monochromatic LE... View full abstract»

• ### Analysis of Particle Movement by Dielectrophoretic Force for Reflective Electronic Display

Publication Year: 2016, Page(s):747 - 752
Cited by:  Papers (2)
| | PDF (551 KB) | HTML

We analyze the movement of particles in an electric paper that consists of upper and lower electrodes. Particles inserted in a fluid move under the influence of an electric field when voltage is applied to the electrodes. The movement of particles is determined by the intensity of the electric field, the viscosity of the fluid, and the q/m of the particles. Each factor has an impact on the indepen... View full abstract»

• ### A High-Efficiency and High-Speed Gain Intervention Refinement Filter for Haze Removal

Publication Year: 2016, Page(s):753 - 759
Cited by:  Papers (3)
| | PDF (2423 KB) | HTML

The dark channel prior has been considered to be an efficient dehazing technique in recent years. However, its operation causes annoying halo effects. To solve this problem, many former imaging filters have been proposed and combined with the dark channel prior operation. However, these filters inevitably induce enormous computational burden while the dehazing effect of the dark channel prior stil... View full abstract»

• ### Algorithm Design for Gamut Mapping From UHDTV to HDTV

Publication Year: 2016, Page(s):760 - 769
| | PDF (3210 KB) | HTML

With the advent of Recommendation ITU-R BT.2020 (Rec. 2020) color space for ultra-high definition television (UHDTV), the development of a gamut mapping algorithm for wide to standard gamuts is greatly needed. In wide-gamut UHDTV production, particularly under the assumption of simultaneous live UHDTV and high-definition television (HDTV) broadcast scenarios, real-time gamut mapping from UHDTV to ... View full abstract»

• ### Blank page

Publication Year: 2016, Page(s): B770
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• ### Introducing IEEE Collabratec

Publication Year: 2016, Page(s): 771
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• ### Journal of Display Technology information for authors

Publication Year: 2016, Page(s): C3
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Publication Year: 2016, Page(s): C4
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## Aims & Scope

This publication covers the theory, design, fabrication, manufacturing and application of information displays and aspects of display technology.

This Journal ceased production in December 2016.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
Arokia Nathan
University of Cambridge
Cambridge, U.K.

format: plain


text:
# [R] Computational Probability

Gabor Grothendieck ggrothendieck at gmail.com
Fri Dec 26 15:03:35 CET 2008

Actually the last line could be simplified to just:

> s > 4 & s < 6
mean   sd sims
[1] 0.72 0.45 2500

On Fri, Dec 26, 2008 at 8:33 AM, Gabor Grothendieck
<ggrothendieck at gmail.com> wrote:
>
>> set.seed(1)
>> library(rv)
>> x <- rvunif(10)
>> s <- simapply(x, sum)
>> mean(s > 4 & s < 6)
>    mean   sd 1% 2.5% 25% 50% 75% 97.5% 99% sims
> [1] 0.72 0.45  0    0   0   1   1     1   1 2500
>
>
>
> On Fri, Dec 26, 2008 at 7:42 AM, Rory Winston <rory.winston at gmail.com> wrote:
>> Hi
>>
>> Firstly , happy Christmas to R-Help! Secondly, I wonder if anyone can help
>> me with the following query: I am trying to reproduce some explicit
>> probability calculations performed in APPL (a Maple extension for
>> computational probability). For instance, in APPL, to compute the
>> probability that the sum of 10 iid uniform variables [0,1] will be between 4
>> and 6, (i..e Pr( 4 < \sum_{i=1}^{10}X_i < 6)), I can type:
>>
>> X := UniformRV(0, 1);
>> Y := ConvolutionIID(X, 10);
>> CDF(Y,6) - CDF(Y,4);
>>
>> which gives the required probability .7222. Is there any way to perform
>> these type of calcuations in R in a general way? I realise that a lot of the
>> machinery behind these computations comes from Maple's symbolic engine, but
>> are there any R extensions for these kind of calculation?
>>
>> Cheers
>> Rory
>>
>>        [[alternative HTML version deleted]]
>>
>> ______________________________________________
>> R-help at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help