worksheet for lab #13
---------------------

a)  read the nature article on randomizing
the deck, "shuffling:  what's the deal?"

b)  quickly read the other web article, make
note of the gray box describing the GSR 
shuffling algorithm.

c)  download the .m files for this week.

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idea
----

studies seem to indicate that 5-7 GSR shuffles
are required to randomize a 52-card deck.  but
can one recognize a particular random deck?  

no.  after all, even 9-1-1 was NY state's random 
winning number on 11 sept 2002.

but one might hope to show that some statistic
of shuffled decks matches the same statistic for
purely random decks.


understand this
---------------

"w13random.m" deals 800 random 52-card decks.

"perfect" is the vector ordered cards 1:52.

"deck" is a vector of the numbers 1-52 in a purely
random order.

a mean-square statistic is calculated which is:

data = sum( (a_j - j)^2 ) / 52

that is, the mean-squared difference between
the random cards & their perfectly ordered
location.

i will show in class tomorrow that the average 
of this mean-square statistic is:

theoretical mean = (Ncards^2 - 1) / 6 = 450.5 (52-cards)

the graph shows the EDF of this mean-square
statistic for the 800 deals.

this .m script has two useful code snippets:
the mean-square statistic loop and an EDF plotter.

be sure you understand what this code does.

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coding exercise
---------------

deal 800 decks using GSR shuffling technique.

i have written a function "shuffle" which 
returns a new random GSR permutation matrix
every time it is called.

try "Pn = shuffle(10);".  then "full(Pn)"
will show the result, and "spy(Pn)" will
make the sparsity plot like in the class
demo.

produce a 8x800 array which stores mean-square
data as 800 deals are made by shuffling a 
perfectly-ordered deck with 8 GSR shuffles.

the pseudocode for the loop looks something
like:

for k=1:Ndeals

  reset deck to perfect
  for j=1:Nshuffles 
     shuffle once
     calculate mean-square stat
  end
end

plot the EDFs which show how the stats 
become more like the random case as the
number of shuffles increases.  remember,
the EDF for a perfectly random deck should
be like the curve of "w13random".