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Example:Finding Probability Distribution

 Post to:   Explanation:

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values.

For a more precise definition one needs to distinguish between discrete and continuous random variables. In the discrete case, one can easily assign a probability to each possible value: when throwing a die, each of the six values 1 to 6 has the probability 1/6. In contrast, when a random variable takes values from a continuum, probabilities are nonzero only if they refer to finite intervals: in quality control one might demand that the probability of a "500 g" package containing between 500 g and 510 g should be no less than 98%.

If total order is defined for the random variable, the cumulative distribution function gives the probability that the random variable is not larger than a given value; it is the antiderivative of the non-cumulative distribution.

## Terminology

As probability theory is used in quite diverse applications, terminology is not uniform and sometimes confusing. The following terms are used for non-cumulative probability distribution functions:

1.  Probability mass, Probability mass function, p.m.f.: for discrete random variables.

2.  Categorical distribution: for discrete random variables with a finite set of values.

3.  Probability density, Probability density function, p.d.f: Most often reserved for continuous random variables.

The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:

1.  Probability distribution function: Continuous or discrete, non-cumulative or cumulative.

2.  Probability function: Even more ambiguous, can mean any of the above, or anything else.

Finally,

1.  Probability distribution: Either the same as probability distribution function. Or understood as something more fundamental underlying an actual mass or density function.

## Discrete probability distribution

A discrete probability distribution shall be understood as a probability distribution 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 as u runs through the set of all possible values of X. It follows that such a random variable can assume only a finite or countably infinite number of values.

In cases more frequently 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 (for example, a distribution over rational numbers). (Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Probability_distribution