What theorem is often expressed as a
Bayes' theorem algorithm and Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry". Bayes' theorem is stated mathematically as the following equation: where formula_2 and formula_3 are events and formula_4. Suppose that a test for using a particular drug is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a
Parseval's theorem "G^" is the integers and this is the case discussed above. When "G" is the real line formula_3, "G^" is also formula_3 and the unitary transform is the Fourier transform on the real line. When "G" is the cyclic group Z, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts. Parseval's theorem can also be expressed as follows: Suppose formula_10 is a square-integrable function over formula_11 (i.e., formula_10 and formula_13 are integrable on that interval), with the Fourier series Then In physics, Parseval's theorem is often written as: where formula_17 represents