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The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
More generally, if two independent random variables X and Y each follow a Cauchy distribution with median equal to zero and shape factor and respectively, then: The ratio distribution for the random variable Z = X / Y {\displaystyle Z=X/Y} is [ 16 ] p Z ( z | a , b ) = a b π 2 ( b 2 z 2 − a 2 ) ln ( b 2 z 2 a 2 ) . {\displaystyle p_{Z}(z ...
The Cauchy distribution (;,) is the distribution of the x-intercept of a ray issuing from (,) with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero
In fact, if a real number x is irrational, then the sequence (x n), whose n-th term is the truncation to n decimal places of the decimal expansion of x, gives a Cauchy sequence of rational numbers with irrational limit x. Irrational numbers certainly exist in , for example:
Pythagorean theorem: It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a , b and the hypotenuse c , sometimes called the Pythagorean equation: [ 6 ]
It is also possible to replace Cauchy sequences in the definition of completeness by Cauchy nets or Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in , then is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces.
A much simpler result, stated in a section above, is that the variance of the product of zero-mean independent samples is equal to the product of their variances. Since the variance of each Normal sample is one, the variance of the product is also one. The product of two Gaussian samples is often confused with the product of two Gaussian PDFs.