Abstract
Probability is a rich and beautiful subject, a discipline unto itself. Its origins were concerned primarily with games of chance, and many lectures on elementary probability theory still contain references to dice, playing cards, and coin flips. These lottery-style scenarios remain useful because they are evocative and easy to understand.
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Notes
- 1.
See Stigler (1986).
- 2.
Its beginning point is usually traced to a text by Jacob Bernoulli, posthumously-published in 1713 (Bernoullli 1713), and its modern endpoint was reached in 1933, with the publication of a text by Kolmogorov (1933).
- 3.
This notation is due to Jeffreys (1931); see his p. 15.
- 4.
For historical comments see Stigler (1986) and Fienberg (2006).
- 5.
We often shorten “probability distribution” to “distribution.” The word distribution is sometimes also applied to data, where it describes the variation among the numbers. However, a probability distribution can refer only to a random variable.
- 6.
In this context terminology is inconsistent: “frequency” can mean either “count” or “relative frequency.”
- 7.
We also get \(\sigma _X= \sqrt{\frac{1}{n}\sum _{i=1}^{n}(x_i - \mu _X)^2}\) which, when we replace \(\mu _X\) with \(\bar{X}\), is not quite the same thing as the sample standard deviation; the latter requires a change from \(n\) to \(n-1\) as the divisor for certain theoretical reasons, including that the sample variance then becomes an unbiased estimator of \(\sigma ^{2}_{X}\). See p. 183.
- 8.
The definitions of expectation and variance assume that the integrals are finite; there are, in fact, some important probability distributions that do not have expectations or variances because the integrals are infinite.
- 9.
Lebesgue integration is a standard topic in mathematical analysis; see for example, Billingsley (1995).
- 10.
The numbers generated by the computer are really pseudo-random numbers because they are created by algorithms that are actually deterministic, so that in very long sequences they repeat and their non-random nature becomes apparent. However, good computer simulation programs use good random number generators, which take an extremely long time to repeat, so this is rarely a practical concern.
- 11.
The cognitive psychology of perception of randomness has been studied quite extensively. See, for instance, Gilovich et al. (1985).
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Kass, R.E., Eden, U.T., Brown, E.N. (2014). Probability and Random Variables. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_3
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