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Encyclopedia > Discrete probability distribution

In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, and X is then called a discrete random variable, if Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

as u runs through the set of all possible values of X.

If a random variable is discrete, then the set of all values that it can assume with nonzero probability is finite or countably infinite, because the sum of uncountably many positive real numbers (which is the smallest upper bound of the set of all finite partial sums) always diverges to infinity. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

In the cases most often considered, this set of possible values is a topologically discrete set in the sense that all its points are isolated points. But there are discrete random variables for which this countable set is dense on the real line. In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an... In topology and related areas of mathematics a subset A of a topological space X is called dense (in X) if the only closed subset of X containing A is X itself. ...

The Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution are among the most well-known discrete probability distributions. In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... In probability and statistics the negative binomial distribution is a discrete probability distribution. ...

Alternative description

Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by jump discontinuities — that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. The points where jumps occur are precisely the values which the random variable may take. The number of such jumps may be finite or countably infinite. The set of locations of such jumps need not be topologically discrete; for example, the cdf might jump at each rational number. In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... Continuous functions are of utmost importance in mathematics and applications. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Representation in terms of indicator functions

For a discrete random variable X, let u0, u1, ... be the values it can assume with non-zero probability. Denote

Omega_i={omega: X(omega)=u_i},, i=0, 1, 2, dots

These are disjoint sets, and by formula (1)

Prleft(bigcup_i Omega_iright)=sum_i Pr(Omega_i)=sum_iPr(X=u_i)=1.

It follows that the probability that X assumes any value except for u0, u1, ... is zero, and thus one can write X as

X=sum_i alpha_i 1_{Omega_i}

except on a set of probability zero, where alpha_i=Pr(X=u_i) and 1A is the indicator function of A. This may serve as an alternative definition of discrete random variables. In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

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