Random variable: Difference between revisions

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Revision as of 18:40, 29 June 2019

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In examples such as these, the [[sample space]] is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.

If <math display = "inline">a_n</math> is an arbitrary set of real numbers, <math display="inline">b_n >0</math> and <math>\sum_n b_n=1</math>, then <math> F=\sum_n b_n \delta_{a_n}</math> is a discrete distribution function. Here <math> \delta_t(x) = 0</math> for <math> x < t</math>, <math> \delta_t(x) = 1</math> for <math> x \ge t</math>. Taking for instance as <math>a_n</math> an enumeration of all rational numbers, one gets a discrete distribution function that is not a step function or piecewise constant.

From a [[Probability axioms|first-principles]]-based approach, a discrete random variable is a random variable whose [[cumulative distribution function]] is [[piecewise constant]].<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/51441829|title=Introduction to Probability|last=Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.|year=|isbn=188652940X|___location=Belmont, Mass.|pages=|oclc=51441829}}</ref>

====Coin toss====