Philip Ackerman <[EMAIL PROTECTED]> wrote:
> Hello,
> I have a question regarding the term "independent, identically
> distributed" random variables. For example, suppose that your random
> variable is height, and the population of interest is adult males. You take
> a sample (size n) of adult males, and measure the variable for each subject.
> Then, if I understand correctly, you have n individual random variables,
> i.e. X1, X2, ..., Xn, which represent the one population random variable X.
> What I do not understand, is why an individual's height is a random variable
> (in addition to height in the population being a random variable), and
> henceforth, has the same distribution as the population random variable. Is
> this because you would get a different height for the same person if you
> took measurements at different times? Does the distribution of height for
> each individual consist of all hypothetical measurements you could take? I
> know that the population distribution of a random variable is also the
> probability distribution when you select one individual randomly from the
> population. Does this have anything to do with "iid?" Any help would be
> greatly appreciated.
A random variable is, by definition, any function whose domain is a
probability space. Both the height of an individual and the average
height of a population are the values of such functions. The *potential*
heights an individual could have form a probability space; the *actual*
height of an individual is the value of a function defined on this space.
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