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The distinction between a random
sample and a realization of a random sample parallels the distinction
between a random variable and a specific realization of that random
variable.
Given a random vector (or random variable) X, a random sample
is a set {X[1], X[2], ...
, X[m]} of random vectors (or random
variables) that are independent and all have the same distribution as
X. Intuitively, we think of the sample as a set of random vectors
(random variables) representing m independent draws from the
distribution of X. A realization of this sample is a set {x[1],
x[2], ... , x[m]}
of vectors (or numbers) all in the range of X. Intuitively, we
think of them as being one result of randomly drawing from the distribution
of X m times. Specifically, if we think of X[k]
as representing the kth random draw from the distribution
of X, then we think of x[k] as
being the result of that random draw.
Suppose random variable X represents the result of tossing a single
6-sided die. A random sample {X[1], X[2],
X[3], X[4], X[5]}
would then represent the result of tossing that die five times. A
realization of that random sample might be {4, 3, 6, 6, 1}.
A random sample is a set of random vectors (random variables). A realization
of a sample is a set of vectors (numbers). |
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