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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]
In probability theory and statistics, the probability distribution of a mixed random variable consists of both discrete and continuous components. A mixed random variable does not have a cumulative distribution function that is discrete or everywhere-continuous. An example of a mixed type random variable is the probability of wait time in a queue.
This random variable X has a Bernoulli distribution with parameter . [29] This is a transformation of discrete random variable. For a distribution function of an absolutely continuous random variable, an absolutely continuous random variable must be constructed.
This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself: = (,).
The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with probability density function f(x) with finite or infinite support on the real line is defined by analogy, using the above form of the entropy as an expectation: [11]: 224
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
The definition extends naturally to more than two random variables. We say that n {\displaystyle n} random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} are i.i.d. if they are independent (see further Independence (probability theory) § More than two random variables ) and identically distributed, i.e. if and only if
A number of special cases are given here. In the simplest case, where the random variable X takes on countably many values (so that its distribution is discrete), the proof is particularly simple, and holds without modification if X is a discrete random vector or even a discrete random element.