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Here, an event is secured when it belongs to a finite sequence of events from the configuration, each of which is enabled by the subset of earlier events from the same sequence. [ 1 ] The nlab simplifies these definitions in two ways:
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]
If A and B are disjoint events, then P(A ∪ B) = P(A) + P(B). This extends to a (finite or countably infinite) sequence of events. However, the probability of the union of an uncountable set of events is not the sum of their probabilities. For example, if Z is a normally distributed random variable, then P(Z = x) is 0 for any x, but P(Z ∈ R ...
For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called events. In this case, {1,3,5} is the event that the die falls on some odd number.
For i ≠ j, the elements q ij are non-negative and describe the rate of the process transitions from state i to state j. The elements q ii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one.
Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain arbitrarily close to , meaning that given a real number greater than zero, all but a finite number of the elements of the sequence have a distance from less than .
The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ΣPr(X n = 0) converges to π 2 /6 ≈ 1.645 < ∞, and so the Borel–Cantelli Lemma states that the set of outcomes that are common to infinitely many such events occurs with probability zero. Hence, the probability of X n = 0 ...
[58] [59] If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. [ 55 ] If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process .