Continuous Random Variables
In probability theory, a continuous random variable is a number that can take any of an infinite number of values, and whose probability density function is defined over an infinite interval. So, its probability density function cannot be expressed as a simple function of a single real number. Continuous random variables are often regarded as being between 0 and 1, but this is not always the case. For example, the probability density function of a normal distribution is defined over the interval [0, 1], and the probability distribution of a normal random variable is a normal distribution. The continuous random variable can be any of a number of different types. The simplest continuous random variable is the real number, but this variable can take any real values. A continuous random variable can be a real number, a complex number, an integer, a rational number, or an irrational number. A continuous random variable can also be a set of real numbers. (See the article on continuous distribution for additional types of continuous random variables.) Continuous random variables are often characterized by their probability density function (pdf), which is often assumed to be continuous. However, in many cases, the assumption of continuous dependence of the pdf on its arguments is not needed, and often not even true. Further, in many cases, the pdf is not continuous, but follows a power law, in which case the assumption of continuous dependence is an approximation. Continuous random variables are often characterized by their probability distribution, which is often assumed to be a normal distribution. However, in many cases, the assumption of a normal probability distribution is not needed, and often not even true. Further, in many cases, the probability distribution is not normal, but follows a power law, in which case the assumption of a normal probability distribution is an approximation.
