On 5/03/2009, at 8:48 PM, Wacek Kusnierczyk wrote:
Rolf Turner wrote:
Sports scores are random variables. You don't know a priori what the
scores are
going to be, do you? (Well, if you do, you must be able to make a
*lot* of money
betting on games!) After the game is over they aren't random any
more; they're
just numbers. But that applies to any random variable. A random
variable is
random only until it is observed, then POOF! it turns into a number.
may i respectfully disagree?
to call for a reference, [1] says (p. 26, def. 1.4.1):
a random variable is a function from sample space S into the real
numbers.
and it's a pretty standard definition.
do you really turn a *function* into a *number* by *observing the
function*? in the example above, you have a sample space, which
consists of possible outcomes of a class of sports events. you have a
random variable -- a function that maps from the number of goals into,
well, the number of goals.
after a sports event, the function is no less random, and no more a
number. you have observed an event, you have computed one realization
of the function (here's your number, which happens to be an
integer) --
but the random variable does not turn to anything.
vQ
[1] Casella, Berger. Statistical Inference, 1st 1990
I was discussing the issue from an elementary/intuitive point of view.
The rigorous mathematical definition of a random variable as a
(measurable)
function from a sample (probability) space is not very helpful to the
beginner.
From the beginner's point of view it is useful to think of random
variables
as being unpredictable quantities that you are *going* to observe.
After
you've observed them, you know what they are and prediction doesn't
come into
it; they are thus no longer random.
From the more mathematical point of view the distinction is between the
function X : Omega |--> R (the real numbers), say, and a *particular
value*
of the function X(omega).
In discussions of statistical inference the viewpoint is always shifting
backwards and forwards between the ``random sample'' X_1, ..., X_n and
the ``realized random sample'' x_1 = X_1(omega), ... x_n = X_n(omega).
Most students --- and I was one of them --- find this shifting point of
view confusing, and I think the elementary heuristic that I introduced
is helpful to many.
cheers,
Rolf
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