Hi Rolf,
>> ... From the beginner's point of view it is useful to think of random
>> variables ...
Who, exactly, is the beginner ? And was not Sir R. A. Fisher pretty arrogant
and fractious ? He also was highly dismissive of Sir Richard Doll's
conclusion that smoking caused cancer (himself being a smoker). Does that
make him a bad statistician, or all statisticians "bad" or arrogant ?
Regards, Mark.
Rolf Turner-3 wrote:
>
>
> 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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