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An expectation E on an algebra A of random variables is a normalized, positive linear functional. What this means is that E[k] = k where k is a constant; E[X * X] ≥ 0 for all random variables X; E[X + Y] = E[X] + E[Y] for all random variables X and Y; and; E[kX] = kE[X] if k is a constant. One may generalize this setup, allowing the algebra ...
If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. Help ... Pages in category "Algebra of random variables"
. the random variables (X n) n∈ all have the same distribution. Note that the random variables of the sequence (X n) n∈ don't need to be independent. The interesting point is to admit some dependence between the random number N of terms and the sequence (X n) n∈.
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
A more general case of this concerns the distribution of the product of a random variable having a beta distribution with a random variable having a gamma distribution: for some cases where the parameters of the two component distributions are related in a certain way, the result is again a gamma distribution but with a changed shape parameter ...
Formally, a multivariate random variable is a column vector = (, …,) (or its transpose, which is a row vector) whose components are random variables on the probability space (,,), where is the sample space, is the sigma-algebra (the collection of all events), and is the probability measure (a function returning each event's probability).
A chart showing a uniform distribution. In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. [1]
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