Random Experiment Event

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A random experiment event is an observable stochastic event that is a subset of some random experiment's sample space (Ω).




  • http://en.wikipedia.org/wiki/Event_%28probability_theory%29#A_note_on_notation
    • Even though events are subsets of some sample space Ω, they are often written as propositional formulas involving random variables. For example, if X is a real-valued random variable defined on the sample space Ω, the event :[math]\displaystyle{ \{\omega\in\Omega \mid u \lt X(\omega) \leq v\}\, }[/math] can be written more conveniently as, simply, :[math]\displaystyle{ u \lt X \leq v\,. }[/math] This is especially common in formulas for a probability, such as: [math]\displaystyle{ P(u \lt X \leq v) = F(v)-F(u)\,. }[/math] The set u < X ≤ v is an example of an inverse image under the mapping X because [math]\displaystyle{ \omega \in X^{-1}((u, v]) }[/math] if and only if [math]\displaystyle{ u \lt X(\omega) \leq v }[/math].



  • (Qian) => Gang Qian. (2008). Basic Probability Theory." Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
    • A event is a subset of the sample space S, a set of samples.
    • Two special events:
      • Certain event: S
      • Impossible or null event: 


  • (Hogg & Ledolter, 1987) ⇒ Robert V. Hogg and Johannes Ledolter. (1987). “Engineering Statistics. Macmillan Publishing Company.
    • Random experiments have outcomes that cannot be determined with certainty before the experiments are performed... The collection of all possible outcomes, namely [math]\displaystyle{ S }[/math] = {H,T}, is called the sample space. Suppose that we are interested in a subset [math]\displaystyle{ A }[/math] of our sample space; for example, in our case, let A={H} represent heads. Repeat this random experiment a number of times, say [math]\displaystyle{ n }[/math], and count the number of times, say [math]\displaystyle{ f }[/math], that the experiment ended in A. Here [math]\displaystyle{ f }[/math] is called the frequency of the event A and the ratio f/n is called the relative frequency of the 'event A in the [math]\displaystyle{ n }[/math] trials of the experiment.


  • (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
    • By an experiment we will mean any procedure that (1) can be repeated, theoretically, an infinite number of times; and (2) has a well-defined set of possible outcomes. Thus, rolling a pair of dice qualifies as an experiment; so does measuring a hypertensive's blood pressure or doing a stereographic analysis to determine the carbon content of moon rocks. Each of the potential eventualities of an experiment is referred to as a sample outcome, [math]\displaystyle{ s }[/math], and their totality is called the sample space, S. To signify the member of [math]\displaystyle{ s }[/math] in [math]\displaystyle{ S }[/math], we write [math]\displaystyle{ s }[/math] In S. Any designated collection of sample outcomes, including individual outcomes, the entire sample space, and the null set, constitutes an event. The latter is said to occur if the outcome of the experiment is one of the members of that event.