According to a textbook I have, a random sample of n objects from a random
variable X, is composed of n random variables itself, namely, X1,X2,...,Xn.
I am having some difficulties in figuring out how to interpret this. For
example, suppose that you are considering the population of adult males in
the U.S., and the random variable is weight. If you take a random sample of
n individuals, are the elements of the sample random (prior to observing
them, of course) because you might observe something different in another
sample due to measurement error? Or perhaps you might get something
different if you took the sample at a different time when weight has
changed? Also, if the elements of a random sample are random variables
themselves, do they have their own parameters, such as mean and standard
deviation, as well as their own density functions and cumulative
distribution functions?
Also, if a statistic is a function of random variables, can a statistic
take the form of a density function with a random vector representing the n
variables? I know, conceptually, that the sampling distribution of a
statistic is purely theoretical and that it represents how a statistic
varies from one sample to another. Mathematically, however, I do not
understand how to represent this, or if the sampling distribution of a
statistic is analogous to the distribution of a random variable which may
have a density function.
I do not know if these questions even make any sense, but the concepts are
fairly confusing to me. Any help would be greatly appreciated.
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