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In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
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]
According to the change-of-variables formula for Lebesgue integration, [21] combined with the law of the unconscious statistician, [22] it follows that [] = for any absolutely continuous random variable X. The above discussion of continuous random variables is thus a special case of the general Lebesgue theory, due to the fact that every ...
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.
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 .
Suppose that X is a random variable whose distribution has a continuous density f. If g is a general function, then the probability that g(X) is valued in a set of real numbers K equals the probability that X is valued in g −1 (K), which is given by ().
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let (,,) be a probability space.Suppose is a random variable with distribution function , and an event on (,,).
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: