Search results
Results from the WOW.Com Content Network
The measurable variables in economics are quantity, quality and distribution. Measuring quantity in economics follows the rules of measuring in physics. Quality as a variable refers to qualitative changes in the production process. Qualitative changes take place when relative of different constant-price input and output factors alter.
Random variables are by definition measurable functions defined on probability spaces. If (,) and (,) are Borel spaces, a measurable function : (,) (,) is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous.
A random variable is a measurable function: from a sample space as a set of possible outcomes to a measurable space.The technical axiomatic definition requires the sample space to be a sample space of a probability triple (,,) (see the measure-theoretic definition).
For example, it is used to equate a probability for a random variable with the Lebesgue-Stieltjes integral typically associated with computing the probability: = for all in the Borel σ-algebra on , where () is the cumulative distribution function for , defined on , while is a probability measure, defined on a σ-algebra of subsets of some ...
If (,,) is a probability space, (,) is a measurable space, and : is a (,)-valued random variable, then the probability distribution of is the pushforward measure of by onto (,). A natural " Lebesgue measure " on the unit circle S 1 (here thought of as a subset of the complex plane C ) may be defined using a push-forward construction and ...
A random variable with values in a measurable space (,) (usually with the Borel sets as measurable subsets) has as probability distribution the pushforward measure X ∗ P on (,): the density of with respect to a reference measure on (,) is the Radon–Nikodym derivative: =.
For example, a sequence of Bernoulli trials is interpreted as the Bernoulli process. One may generalize this to include continuous time Lévy processes, and many Lévy processes can be seen as limits of i.i.d. variables—for instance, the Wiener process is the limit of the Bernoulli process.
For a fixed measurable function , is a random variable with mean and variance (). By the strong law of large numbers , P n ( A ) converges to P ( A ) almost surely for fixed A . Similarly P n f {\displaystyle P_{n}f} converges to E f {\displaystyle \mathbb {E} f} almost surely for a fixed measurable function f {\displaystyle f} .