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For a random variable following the continuous uniform distribution, the expected value is = +, and the variance is = (). For the special case a = − b , {\displaystyle a=-b,} the probability density function of the continuous uniform distribution is:
Indeed, even when the random variable does not have a density, the characteristic function may be seen as the Fourier transform of the measure corresponding to the random variable. Another related concept is the representation of probability distributions as elements of a reproducing kernel Hilbert space via the kernel embedding of distributions .
A mixed random variable is a random variable whose cumulative distribution function is neither discrete nor everywhere-continuous. [10] It can be realized as a mixture of a discrete random variable and a continuous random variable; in which case the CDF will be the weighted average of the CDFs of the component variables. [10]
An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous. There are many examples of absolutely continuous probability distributions: normal , uniform , chi-squared , and others .
A chart showing a uniform distribution. In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. [1]
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to random variables having a standard uniform distribution. [1]
If is a continuous random variable with probability density function () , then = + is a ... Uniform distribution (continuous) Uniform distribution (discrete)
In the theory of probability, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables., [5] [6] [7] that is, 1. A class of random variables is called uniformly integrable if: