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If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem. If the characteristic function φ X of a random variable X is integrable, then F X is absolutely continuous, and therefore X has a probability density function.
The characteristic function of a cooperative game in game theory. The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics. The Euler characteristic, a topological invariant. The receiver operating characteristic in statistical decision theory. The point characteristic function in statistics.
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 ...
If the characteristic function of some random variable is of the form () = in a neighborhood of zero, where () is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that can be at most a quadratic polynomial, and therefore is a normal random variable. [31]
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.
Characteristics are also a powerful tool for gaining qualitative insight into a PDE. One can use the crossings of the characteristics to find shock waves for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to along itself. Thus, when two characteristics cross, the function ...
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The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative.