While for a discrete random variable one could say that an event with probability zero is impossible, this can not be said in the case of a continuous random variable, because then no value would be possible.
This paradox is solved by realizing that the probability that X attains a value in an uncountable set (for example an interval) can not be found by adding the probabilities for individual values.
By another convention, the term "continuous random variable" is reserved for random variables that have probability density functions. A random variable with the Cantor distribution is continuous according to the first convention, and according to the second, is neither continuous nor discrete nor a weighted average of continuous and discrete random variables.
In practical applications random variables are often either discrete or continuous.
While for a discrete randomvariable one could say that an event with probability zero is impossible, this can not be said in the case of a continuousrandomvariable, because then no value would be possible.
By another convention, the term "continuousrandomvariable" is reserved for randomvariables that have probability density functions.
A randomvariable with the Cantor distribution is continuous according to the first convention, and according to the second, is neither continuous nor discrete nor a weighted average of continuous and discrete randomvariables.
A randomvariable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result.
Unlike the common practice with other mathematical variables, a randomvariable cannot be assigned a value; a randomvariable does not describe the actual outcome of a particular experiment, but rather describes the possible, as-yet-undetermined outcomes in terms of real numbers.
Mathematically, a randomvariable is defined as a measurable function from a probability space to some measurable space.
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