Ignoring the domain of RV in punif()

Before the ticket finally enters the waste bin, I think it is
necessary to explicitly explain what is meant by the "domain"
of a random variable. This is not (though in special cases
could be) the space of possible values of the random variable.

Definition of (real-valued) Random Variable (RV):
Let Z be a probability space, i.e. a set {z} of entities z
on which a probability distribution is defined. The entities z
do not need to be numeric. A real-valued RV X is a function
X:Z --> R defined on Z such that, for any z in Z, X(z) is a
real number. The set Z, in tthis context, is (by definitipon)
the *domain* of X, i.e. the space on which X is defined.
It may or may not be (and usually is not) the same as the set
of possible values of X.

Then. given any real value x0, the CDF of X at x- is Prob[X <= X0].
The distribution function of X does not define the domain of X.

As a simple exam[ple: Suppose Q is a cube of side A, consisting of
points z=(u,v,w) with 0 <= u,v,w <= A. Z is the probability space
of points z with a uniform distribution of position within Q.
Define the random variable X:Q --> [0,1] as
  X(u,v,w) = x/A
Then X is uniformly distributed on [0,1], the domain of X is Q.
Then for x <= 0 _Prob[X <= x] = 0, for 0 <= x <= 1 Prob(X >=x] = x,
for x >= 1 Prob(X <= x] = 1. These define the CDF. The set of poaaible
values of X is 1-dimensional, and is not the same as the domain of X,
which is 3-dimensional.

Hopiong this helps!
Ted.

Ignoring the domain of RV in punif()

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