# Probability

• John R. Klauder
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

Flip a coin, throw some dice, measure the voltage of a noisy circuit, and the outcomes are realizations of random variables. Let X denote the random variable in question, which we assume takes on strictly real values, and let J denote the set of values X can assume. The set J may be discrete (as for dice), continuous (as for noisy voltages), or the union of the two. Repeated and independent measurements yield a sequence of possible values x n , where$$n = 1, 2, \ldots , N, N < \infty,$$ that sample the possible outcomes of X in an unbiased way. Approximate averages involving the x n values can be obtained easily, such as
$$\overline{X} = \frac{1}{N} \sum^{N}_{n=1} x_n,$$
$$\overline{{X^2}} = \frac{1}{N} \sum^{N}_{n=1} x^{2}_n,$$
etc. Experience shows that the larger N becomes, the closer these values generally are to those values of the true averages for these quantities obtained ideally when $$N \rightarrow \infty$$

## Keywords

Probability Measure Characteristic Function Central Limit Theorem Divisible Distribution Poisson Variable
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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