# Complex numbers probability theory

• ## Can a probability be a complex number?

According to the author of various papers on exotic probability, Saul Youssef, the valid possible alternatives for probability values are the real numbers, the complex numbers and the quaternions..

• ## Can probability be a complex number?

Indeed, (in mathematics) a probability measure takes values in [0,1] by definition.
Probabilities were defined to be positive valued for the same reason why lengths, areas and volumes are positive valued.
We can generalize it to be complex, vector or even matrix(operator) valued.Jun 26, 2015.

• ## What is a complex number in statistics?

Complex numbers is a kind of two-dimensional vectors whose components are the so-called real part and imaginary part.
The real part usually corresponds to physical quantities while the imaginary part is a purely mathematical construction..

• ## What is a complex probability?

Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities..

• ## What is the application of complex numbers in statistics?

There are two broad classes of use of complex numbers in statistics, one being when the underlying problem uses complex numbers (leading to complex random variables), and the other being when tools using complex numbers are used to describe statistical problems involving only real random variables.Feb 24, 2019.

• ## What is the reason for complex numbers?

Complex numbers arise from the need to express negative numbers' roots, which real numbers can't do.
This is why they reflect all the roots of polynomials.
Their use extends to different scientific branches, ranging from mathematics to engineering..

• ## What is the use of complex numbers in statistics?

There are two broad classes of use of complex numbers in statistics, one being when the underlying problem uses complex numbers (leading to complex random variables), and the other being when tools using complex numbers are used to describe statistical problems involving only real random variables.Feb 24, 2019.

• ## Where are complex numbers used?

Their use extends to different scientific branches, ranging from mathematics to engineering.
Complex numbers can also represent electromagnetic waves and electric currents, so they are essential in the field of electronics and telecommunications..

• One reason is that complex numbers represent direction algebraically (.
1. D vectors) and many of their operations have a direct geometrical meaning (e
2. .g., the product rule: multiply the norms and add the angles).
• The complex number is in the form of a+ib, where a = real number and ib = imaginary number.
Also, a,b belongs to real numbers and i = √-1.
Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number.
• This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna, though his understanding was rudimentary; moreover he later described complex numbers as "as subtle as they are useless".
In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible  ExamplesProbability density functionExpectationCovariance and pseudo

Fourier transform of the probability density function

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution.
If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function.
Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.
There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.
Exotic probability is a branch of probability theory that deals with probabilities which are outside the normal range of [0, 1].

Overview of and topical guide to probability

Probability is a measure of the likeliness that an event will occur.
Probability is used to quantify an attitude of mind towards some proposition whose truth is not certain.
The proposition of interest is usually of the form A specific event will occur. The attitude of mind is of the form How certain is it that the event will occur? The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 is called the probability.
Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

Power series derived from a discrete probability distribution

In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable.
Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.

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