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The characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite. A characteristic function is uniformly continuous on the entire space. It is non-vanishing in a region around zero: φ(0) = 1. It is bounded: | φ(t) | ≤ 1.
In mathematics, the term "characteristic function" can refer to any of several distinct concepts: The indicator function of a subset , that is the function 1 A : X → { 0 , 1 } , {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},} which for a given subset A of X , has value 1 at points of A and 0 at points of X − A .
Therefore, y′ = re rx, y″ = r 2 e rx, and y (n) = r n e rx are all multiples. This suggests that certain values of r will allow multiples of e rx to sum to zero, thus solving the homogeneous differential equation. [5] In order to solve for r, one can substitute y = e rx and its derivatives into the differential equation to get
This is the characteristic function of the normal distribution with expected value + and variance + Finally, recall that no two distinct distributions can both have the same characteristic function, so the distribution of X + Y must be just this normal distribution.
In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function , and one can freely convert between the two, but the characteristic function as defined below is better ...
When μ = 0, the distribution of Y is a half-normal distribution. The random variable (Y/σ) 2 has a noncentral chi-squared distribution with 1 degree of freedom and noncentrality equal to (μ/σ) 2. The folded normal distribution can also be seen as the limit of the folded non-standardized t distribution as the degrees of freedom go to infinity.
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In classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). In fuzzy set theory, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in some algebra or structure (usually required to be at least a poset or lattice).