| Organodynamics | Grant
Holland, Apr 25, 2014 |
| Slide: Random Variables | |
| Loosely defined, Òrandom variableÓ means:
A variable that represents the set of values that a random trial can realizeÉ i.e., a sample space. This
idea enables us to use a variable symbol (e.g. X) within mathematical expressions in a manner similar to how we use ÒpureÓ variables. [Reference to the author of Stochastic processes in Physics] However,
there is a stricter definition, and more broadly adopted one, which implies that a random variable isÉ Òa collection of numerical values such as temperature, voltages, numbers of arrivalsÉÓ [Gallager 2011] Random
variable: ÒÉa function
X from the sample space of a probability [space] to the set of real numbers R.Ó [Gallager 2011] | Random
Variable In other words, the strict definition of random variable is the value
function v(X) that we introduced in the previous slide. There is an unfortunate ambiguity across these two definition: The loose usage of random variable applies to all probability spaces. While the stricter definition applies only to probability spaces for which a value function is defined. This strict usage of random variable to require a mapping of a probability space to R is the most frequent. So, the strict definition of random variable does
not apply to organodynamic probability spaces (OSS)! |
| Unfortunately,
both the loose and strict uses of Òrandom variableÓ are intermingled, conflated, and thrown around quite causally in the annals of probability. This
ambiguous usage results in confusion within the language of organodynamics. I propose the term Òchance variableÓ to be used in place of the loose usage of Òrandom variableÓ. Thus, the term Òrandom variableÓ will be restricted to its stricture usage Ð which implies Òthe mapping of the sample space to real numbers. | According
to this usage: Organodynamics uses chance variables, but not necessarily random variables.
(The mathematics of organodynamics must be usfficiently general so that it cannot assume that there is a mapping of the sample space to the real numbers.) |
Notes: