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In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders , so that one random variable A {\displaystyle A} may be neither stochastically greater than, less than, nor equal to another random variable B {\displaystyle B} .
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
Random variables are usually written in upper case Roman letters, such as or and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable.
Stochastic dominance is a partial order between random variables. [1] [2] It is a form of stochastic ordering.The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers.
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, [1] Edmund Landau, [2] and others, collectively called Bachmann–Landau notation or asymptotic notation.
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. [1] The term 'random variable' in its mathematical definition refers to neither randomness nor variability [ 2 ] but instead is a mathematical function in which
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.