FINGER EXERCISES:  [discuss at least some Fri Sept 13]

1. Many pseudo-random number generators on the computer can be though of
as "function iteration schemes":  Starting with a seed x0, the random
number generator generates successive numbers x1, x2, x3, .. as
follows:

    x1 = f(x0)
    x2 = f(x1)
    x3 = f(x2)

    etc
    
where f(x) is some numerical function defined by the code for the random
number generator.  (The idea of designing a good random number generator is
to find a function f() such that the sequence x0 x1 x2 ... passes a
variety of statistical tests for "randomness").

Prove rigorously that any actual pseudo random number generator of this
form can produce only a finite sequence of x's before repeating itself.
State and justify any reasonable assumptions you make in your proof.

2. Prove that any computer program, no matter how complicated or on what
digital computer, can produce only a finite number of
different outputs.  State and justify any reasonable assumptions you make
in your proof.

3. Suppose dataframe "mydata" has m columns of numerical data.  Write a
function "histos" in S, that takes "mydata" as an argument and produces
an approximately square array of histogram plots in a single motif() or
x11() window.  You don't know m in advance, so you have to figure out,
inside your function, what "square array of histogram plots" means, for
each possible m.  Demonstrate that your function works with a few
different made-up data frames.

[to make up a dataframe: 
          mydata _ as.data.frame(matrix(rnorm(I*J),ncol=J))
]

4. "Machine epsilon" is the smallest number e such that 1+e>1.  Write
some lines of code in S (possibly but not necessarily in a function) that
explores for and finds machine epsilon for S, by adding several small
values to 1 and seeing if the result equals 1.