A new collection has been released on Maple Learn! The new Pascal’s Triangle Collection allows students of all levels to explore this simple, yet widely applicable array.

Though the binomial coefficient triangle is often referred to as Pascal’s Triangle after the 17th-century mathematician Blaise Pascal, the first drawings of the triangle are much older. This makes assigning credit for the creation of the triangle to a single mathematician all but impossible.

Persian mathematicians like Al-Karaji were familiar with the triangular array as early as the 10th century. In the 11th century, Omar Khayyam studied the triangle and popularised its use throughout the Arab world, which is why it is known as “Khayyam’s Triangle” in the region. Meanwhile in China, mathematician Jia Xian drew the triangle to 9 rows, using rod numerals. Two centuries later, in the 13th century, Yang Hui introduced the triangle to greater Chinese society as “Yang Hui’s Triangle”. In Europe, various mathematicians published representations of the triangle between the 13th and 16th centuries, one of which being Niccolo Fontana Tartaglia, who propagated the triangle in Italy, where it is known as “Tartaglia’s Triangle”. 

Blaise Pascal had no association with the triangle until years after his 1662 death, when his book, Treatise on Arithmetical Triangle, which compiled various results about the triangle, was published. In fact, the triangle was not named after Pascal until several decades later, when it was dubbed so by Pierre Remond de Montmort in 1703.

The Maple Learn collection provides opportunities for students to discover the construction, properties, and applications of Pascal’s Triangle. Furthermore, students can use the triangle to detect patterns and deduce identities like Pascal’s Rule and The Binomial Symmetry Rule. For example, did you know that colour-coding the even and odd numbers in Pascal’s Triangle reveals an approximation of Sierpinski’s Fractal Triangle?

See Pascal’s Triangle and Fractals

Or that taking the sum of the diagonals in Pascal's Triangle produces the Fibonacci Sequence?

See Pascal’s Triangle and the Fibonacci Sequence

Learn more about these properties and discover others with the Pascal’s Triangle Collection on Maple Learn. Once you are confident in your knowledge of Pascal’s Triangle, test your skills with the interactive Pascal’s Triangle Activity. 

For years I've been angry that Maple isn't capable of formally manipulating random vectors (aka multivariate random variables).
For the record Mathematica does.

The problem I'm concerned with is to create a vector W such that

type(W, RandomVariable)

will return true.
Of course defining W from its components w₁, .., w_N, where each w_n  is a random variable is easy, even if these components are correlated or, more generally dependent ( the two concepts being equivalent iif all the w_n  are gaussian random variables).
But one looses the property that W is no longer a (multivariate) random variable.
See a simple example here: NoRandomVectorsInMaple.mw

This is the reason why I've developped among years several pieces of code to build a few multivariate random variable (multinormal, Dirichlet, Logistic-Normal, Skew Multivariate Normal, ...).

In the framework of my activities, they are of great interest and the purpose of this post is to share what I have done on this subject by presenting the most classic example: the multivariate gaussian random variable.

My leading idea was (is) to build a package named MVStatistics on the image of the Statistics package but devoted to Multi Variate random variables.
I have already construct such a package aggregating about fifty different procedures. But this latter doesn't merit the appellation of "Maple package" because I'm not qualified to write something like this which would be at the same time perennial, robust, documented, open and conflict-free with the  Statistics package.
In case any of you are interested in pursuing this work (because I'm about to change jobs), I can provide it all the different procedures I built to construct and manipulate multivariate random variables.

To help you understand the principles I used, here is the most iconic example of a multivariate gaussian random variable.
The attached file contains the following procedures

MVNormal
  Constructs a gaussian random vector whose components can be mutually correlated
  The statistics defined in Distribution are: (this list could be extended to other
  statistics, provided they are "recognized" statitics, see at the end of this 
  post):
      PDF
      Mode
      Mean
      Variance
      StandardDeviation = add(s[k]*x[k], k=1..K)
      RandomSample

DispersionEllipse
  Builds and draws the dispersion ellipses of a bivariate gaussia, random vector

DispersionEllipsoid
  Builds and draws the dispersion ellipsoids of a trivariate gaussia, random vector

MVstat
  Computes several statistics of a random vector (Mean, Variance, ...)

Iserlis
  Computes the moments of any order of a gaussian random vector

MVCentralMoment
  Computes the central moments of a gaussian random vector

Conditional
  Builds the conditional random vector of a gaussian random vector wrt some of its components 
  the moments of any order of a gaussian random vector.
  Note: the result has type RandomVariable.

MarginalizeAgainst
  Builds the marginal random vector of a gaussian random vector wrt some of its components 
  the moments of any order of a gaussian random vector.
  Note: the result has type RandomVariable.

MardiaNormalityTest
  The multi-dimensional analogue of the Shapiro-Wilks normality test

HZNormalityTest
  Henze-Zirkler test for Multivariate Normality

MVWaldWolfowitzTest
  A multivariate version of the non-parametrix Wald-Folfowitz test

Do not hesitate to ask me any questions that might come to mind.
In particular, as Maple introduces limitations on the type of some attributes (for instance Mean  must be of algebraic type), I've been forced to lure it by transforming vector or matrix quantities into algebraic ones.
An example is

Mean = add(m[k]*x[k], k=1..K)

where m[k] is the expectation of the kth component of this random vector.
This implies using the procedure MVstat to "decode", for instance, what Mean returns and write it as a vector.

MultivariateNormal.mw

About the  statistics ths Statistics:-Distribution constructor recognizes:
To get them one can do this (the Normal distribution seems to be the continuous one with the most exhaustive list os statistics):

restart
with(Statistics):
X := RandomVariable(Normal(a, b)):
attributes(X);
      protected, RandomVariable, _ProbabilityDistribution

map(e -> printf("%a\n", e), [exports(attributes(X)[3])]):
Conditions
ParentName
Parameters
CharacteristicFunction
CDF
CGF
HodgesLehmann
Mean
Median
MGF
Mode
PDF
RousseeuwCrouxSn
StandardDeviation
Support
Variance
CDFNumeric
QuantileNumeric
RandomSample
RandomSampleSetup
RandomVariate
MaximumLikelihoodEstimate

Unfortunately it happens that for some unknown reason a few statistics cannot be set by the user.
This is for instance the case of Parameters serious consequences in certain situations.
Among the other statistics that cannot be set by the user one finds:

• ParentName,
• QuantileNumeric  whose role is not very clear, at least for me, but which I suspect is a procedure which "inverts" the CDF to give a numerical estimation of a quantile given its probability.
  If it is so accessing  QuantileNumeric would be of great interest for distributions whose the quantiles have no closed form expressions.
• CDFNumeric  (same remark as above)

Finally, the statistics Conditions, which enables defining the conditions the elements of Parameters must verify are not at all suited for multivariate random variables.
It is for instance impossible to declare that the variance matrix (or the correlation matrix) is a square symmetric positive definite matrix).