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
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