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.
In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers.
While real probability density functions have always real positive values, the probability density functions of imaginary random variables are found to be always imaginary negative.
Complex numbers are the combination of real and imaginary numbers. The real part can be expressed by an integer or decimal, while the imaginary part has a square that is negative. Complex numbers arise from the need to express negative numbers' roots, which real numbers can't do.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a+bi where a is the real part and bi is the imaginary part. For example, 5+2i is a complex number. So, too, is 3+4√3i.
The complex number system is an extension of the real number system. All real numbers are complex numbers (by having b = 0), but there are complex numbers that are not real, e.g. 2 - 3i. We add or subtract two complex numbers by adding and substracting the corre- sponding real and complex part of the number.
If I set t equal to 0, then the probability density at X=0, for instance becomes: 1(0)(√2π)e(00)2, that is, as X tends to 0, the probability density tends to infinity.
It becomes zero at radial nodes. In 2s graph, the probability density curve becomes zero at a point which is know as node.
It's a well-known fact that the largest value a probability can take is 1. However, for some PDFs (e.g. the PDF of the exponential distribution, the graph below), when λ= 1.5 and 𝒙 = 0, the probability density is 1.5, which is obviously greater than 1!
This means a probability number is always a number from 0 to 1. Probability can also be written as a percentage, which is a number from 0 to 100 percent. The higher the probability number or percentage of an event, the more likely is it that the event will occur.
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.
Compound Probability Formulas
= P (A) + P(B). For mutually inclusive events, P (A or B) = P(A) + P(B) - P(A and B). Using the organized list method, you would list all the different possible outcomes that could occur.
Types of Probability
There are three major types of probabilities: Theoretical Probability. Experimental Probability. Axiomatic Probability.
What are the types of probability? Probability is the branch of mathematics concerning the occurrence of a random event, and four main types of probability exist: classical, empirical, subjective and axiomatic.
Specifically, it is assumed that an event with exactly zero probability (that does not approximate an infinitesimal value) can have strictly positive probabilities. This means that such an event can be possible which implies that its zero probability does not mean impossibility.
The experimental probability of an event cannot be greater than 1 since the number of trials in which the event can happen cannot be greater than the total number of trials.
Probability is a measure of the likeliness or chance of possibilities in a situation. It is not possible to have a length or area or weight or height less than zero. In the same way, probability cannot be below 0. You can't have less than zero chance of something happening.
Probabilities have no unit, must be numbers between zero and one, and the total probability must equal one. The position probability density in one dimension has unit m−1 (“probability per unit length”) and can in general have a numerical value that is greater than one.
The probability density (or probably distribution) is given by taking the square of the absolute value of the wave function. It gives us the likelihood of finding an electron (or some other system) at some given point in space.
In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables.
A complex number is a number of the form a + bi, where a, b are real "coefficients" called the real and imaginary part. Just like n often represents an integer, z often represents a complex number. All real numbers (like 0.5, √3, π, ...) are complex numbers, as are all imaginary numbers (like 0.5i, i√3, πi, ...).
The short answer is this: Yes, the complex numbers are closed for polynomials using the algebraic operations of addition, multiplication, subtraction, and division. Yes, there are larger domains than the complex numbers, such as the quaternions and octonians.
Conclusion. So yes, there are more complex number systems than the complex numbers, one of which we saw here in the form of quaternions—the 4-dimensional numbers.