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Statistical Distribution


Statistical Distribution

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.[1][2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).[3]

For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). 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.[4]

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by Ω \displaystyle \Omega , is the set of all possible outcomes of a random phenomenon being observed; it may be any set: a set of real numbers, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip would be Ω = heads, tails.

To define probability distributions for the specific case of random variables (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p \displaystyle p assigning a probability to each possible outcome: for example, when throwing a fair dice, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is p ( 2 ) + p ( 4 ) + p ( 6 ) = 1 / 6 + 1 / 6 + 1 / 6 = 1 / 2. \displaystyle p(2)+p(4)+p(6)=1/6+1/6+1/6=1/2.

Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval.[5] An alternative description of the distribution is by means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., P ( X

Probability distributions usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function. On the other hand, absolutely continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In the absolutely continuous case, probabilities are described by a probability density function, and the probability distribution is by definition the integral of the probability density function.[4][5][8] The normal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.[10]

Any probability distribution can


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