Mathematical Description of Random Sample
In mathematical terms, given a random variable X with distribution F, a random sample of length n (where n may be any of 1,2,3,...) is a set of n independent, identically distributed (iid) random variables with distribution F.
A sample concretely represents n experiments in which the same quantity is measured. For example, if X represents the height of an individual and n individuals are measured, will be the height of the i-th individual. Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these variables (which are the values that these random variables take, formally called random variates). In other words, is a function representing the measurement at the i-th experiment and is the value actually obtained when making the measurement.
The concept of a sample thus includes the process of how the data are obtained (that is, the random variables). This is necessary so that mathematical statements can be made about the sample and statistics computed from it, such as the sample mean and covariance.
Read more about this topic: Sample (statistics)
Famous quotes containing the words mathematical, description, random and/or sample:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“I was here first introduced to Joe.... He was a good-looking Indian, twenty-four years old, apparently of unmixed blood, short and stout, with a broad face and reddish complexion, and eyes, methinks, narrower and more turned up at the outer corners than ours, answering to the description of his race. Besides his underclothing, he wore a red flannel shirt, woolen pants, and a black Kossuth hat, the ordinary dress of the lumberman, and, to a considerable extent, of the Penobscot Indian.”
—Henry David Thoreau (1817–1862)
“Assemble, first, all casual bits and scraps
That may shake down into a world perhaps;
People this world, by chance created so,
With random persons whom you do not know—”
—Robert Graves (1895–1985)
“The present war having so long cut off all communication with Great-Britain, we are not able to make a fair estimate of the state of science in that country. The spirit in which she wages war is the only sample before our eyes, and that does not seem the legitimate offspring either of science or of civilization.”
—Thomas Jefferson (1743–1826)