Events in Probability Spaces
Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution, the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers 'badly-behaved' sets, such as those that are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice.
In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the σ-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the σ-algebra.
Read more about this topic: Event (probability Theory)
Famous quotes containing the words events in, events, probability and/or spaces:
“Whatever events in progress shall disgust men with cities, and infuse into them the passion for country life, and country pleasures, will render a service to the whole face of this continent, and will further the most poetic of all the occupations of real life, the bringing out by art the native but hidden graces of the landscape.”
—Ralph Waldo Emerson (1803–1882)
“All strange and terrible events are welcome,
But comforts we despise.”
—William Shakespeare (1564–1616)
“Crushed to earth and rising again is an author’s gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.”
—Fannie Hurst (1889–1968)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)