quality:corruption, stylistic:style_transfer
Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often represented in notation by Ω , {\displaystyle \ \Omega \,} is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set: a set of real numbers, a set of descriptive labels, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could 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 (e.g. when throwing a fair die, each of the six digits “1” to “6”, corresponding to the number of dots on the die, has the probability 1 6). {\displaystyle \ {\tfrac {1}{6}}~).} The probability of an event is then defined to be the sum of the probabilities of all 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({\text{“}}2{\text{”}})+p({\text{“}}4{\text{”}})+p({\text{“}}6{\text{”}})={\tfrac {1}{6}}+{\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{2}}~.}

In contrast, when a random variable takes values from a continuum then by convention, any individual outcome is assigned probability zero. For such continuous random variables, only events that include infinitely many outcomes such as intervals have probability greater than 0.

For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then, the probability that it weighs exactly 500 g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero decimal digits in the remaining omitted digits ignored by the precision level.

However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability. This is possible because this measurement does not require as much precision from the underlying equipment.

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. 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 < x) {\displaystyle \ {\boldsymbol {\mathcal {P}}}(X<x)\ } for some x {\displaystyle \ x\ }). The cumulative distribution function is the area under the probability density function from − ∞ {\displaystyle \ -\infty \ } to x , {\displaystyle \ x\,} as shown in figure 1.