WEAK LAWS OF LARGE NUMBERS FOR ARRAYS UNDER A
... In the following let {un , n ≥ 1} and {vn , n ≥ 1} be two sequences of integers (not necessary positive or finite) such that vn > un for all n ≥ 1 and vn − un → ∞ as n → ∞. Let {kn , n ≥ 1} be a sequence of positive numbers such that kn → ∞ as n → ∞. Definition 1.1. Let {Xni , un ≤ i ≤ vn , n ≥ 1} b ...
... In the following let {un , n ≥ 1} and {vn , n ≥ 1} be two sequences of integers (not necessary positive or finite) such that vn > un for all n ≥ 1 and vn − un → ∞ as n → ∞. Let {kn , n ≥ 1} be a sequence of positive numbers such that kn → ∞ as n → ∞. Definition 1.1. Let {Xni , un ≤ i ≤ vn , n ≥ 1} b ...
probability
... Example: Suppose a certain company insists that it complies with “gender equality” regulations among its employee population, i.e., approx. 50% male and 50% female. To test this claim, let us select a random sample of n = 100 employees, and count X = the number of males. (If the claim is true, then ...
... Example: Suppose a certain company insists that it complies with “gender equality” regulations among its employee population, i.e., approx. 50% male and 50% female. To test this claim, let us select a random sample of n = 100 employees, and count X = the number of males. (If the claim is true, then ...
Probabilities Involving “and”, “or”, “not”
... liked to go fishing, and 12 said they don’t enjoy either activity. How many enjoy fishing but not sailing? ...
... liked to go fishing, and 12 said they don’t enjoy either activity. How many enjoy fishing but not sailing? ...
Random Variables. . . in a Nutshell
... In principle, there are infinitely many different probability mass functions. In practice, there are a few families of parametrized probability mass functions which typically suffice to “model” most random phenomena — most experiments — of interest; we have presented here two of the more common, uniform ...
... In principle, there are infinitely many different probability mass functions. In practice, there are a few families of parametrized probability mass functions which typically suffice to “model” most random phenomena — most experiments — of interest; we have presented here two of the more common, uniform ...
A SURVEY OF AVERAGE CONTRACTIVE ITERATED FUNCTION
... Basic courses in finite state space Markov chains traditionally lack discussions of representations of the form (1) when proving convergence results with the unfortunate consequence that connections to basic time-series models, like e.g. the autoregressive moving average model, appear vague. Hopeful ...
... Basic courses in finite state space Markov chains traditionally lack discussions of representations of the form (1) when proving convergence results with the unfortunate consequence that connections to basic time-series models, like e.g. the autoregressive moving average model, appear vague. Hopeful ...
Randomness
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or ""trials"") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators.Random selection is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.