Singular distribution
Encyclopedia
In probability
, a singular distribution is a probability distribution
concentrated on a set of Lebesgue measure zero
, where the probability of each point in that set is zero. These distributions are sometimes called singular continuous distributions. Such distributions are not absolutely continuous with respect to Lebesgue measure
.
A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function
, since the Lebesgue integral of any such function would be zero.
An example is the Cantor distribution.
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, a singular distribution is a probability distribution
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....
concentrated on a set of Lebesgue measure zero
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
, where the probability of each point in that set is zero. These distributions are sometimes called singular continuous distributions. Such distributions are not absolutely continuous with respect to Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
.
A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
, since the Lebesgue integral of any such function would be zero.
An example is the Cantor distribution.