Wiktionary
n. (context mathematics English) A function of a discrete random variable yielding the probability that the variable will have a given value
Wikipedia
In probability and statistics, a probability distribution is a mathematical description of a random phenomenon in terms of the probabilities of events. Examples of random phenomena include the results of an experiment or survey. A probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a higher-dimensional vector space, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be {Heads, Tails}. Probability distributions are generally divided into two classes. A discrete probability distribution can be encoded by a list of the probabilities of the outcomes, known as a probability mass function. On the other hand, in a continuous probability distribution, the probability of any individual outcome is 0. Continuous probability distributions can often be described by probability density functions; however, more complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.
In applied probability, a probability distribution can be specified in a number of different ways, often chosen for mathematical convenience:
- by supplying a valid probability mass function or probability density function
- by supplying a valid cumulative distribution function or survival function
- by supplying a valid hazard function
- by supplying a valid characteristic function
- by supplying a rule for constructing a new random variable from other random variables whose joint probability distribution is known.
A probability distribution whose sample space is the set of real numbers is called univariate, while a distribution whose sample space is a vector space is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a list of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.
Usage examples of "probability distribution".
A red dragonfly hovers above a backwater of the stream, its wings moving so fast that the eye sees not wings in movement but a probability distribution of where the wings might be, like electron orbitals: a quantum-mechanical effect that maybe explains why the insect can apparently teleport from one place to another, disappearing from one point and reappearing a couple of meters away, without seeming to pass through the space in between.
In effect, Tegmark is asking us to accept a general principle: that whenever you have a phase space (statisticians would say a sample space) with a well-defined probability distribution, then everything in that phase space must be real.
The potentiality of all outcomes remains up until that moment, and none of them can be absolutely ruled out, but a weighted probability distribution can be assigned to them, which allows the effective prediction of which single event will actually occur.