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In probability theory and statistics, a probability distribution is a mathematical function that gives the probabilities of the occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events. For instance, if it is used to denote the outcome of a coin toss, then the probability distribution would take the values 0.5 for and 0.5 for. Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. General definition A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies to absolutely continuous and discrete variables, is by means of a probability function P\colon \mathcal \int_a^b d x\, \Psi ^2, the probability that the particle's position will be in the interval in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics. Probabilistic load flow in the power-flow study explains the uncertainties of input variables as a probability distribution and provides the power flow calculation also in terms of a probability distribution. Prediction of natural phenomena occurrences based on previous frequency distributions such as tropical cyclones, hail, time in between events, etc. 

See also Conditional probability distribution 

Joint probability distribution 

 Quasiprobability distribution 

 Empirical probability distribution 

 Histogram Riemann–Stieltjes integral application to probability theory list

 List of probability distributions 

 List of statistical topics