In terms of random variables, how is an exponential random variable commonly defined?

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An exponential random variable is defined as a continuous random variable, which means that it can take on an infinite number of values over a given range. The exponential distribution is primarily used to model the time until an event occurs, such as the time between arrivals of customers in a queue or the time until a radioactive particle decays.

The probability density function (PDF) of an exponential random variable is defined for all non-negative values, reflecting its nature as a continuous distribution. This unique property distinguishes it from discrete random variables, which can only take on specific values, typically integers. In essence, the exponential random variable allows for calculations related to continuous outcomes and their probabilities over intervals, making it a vital concept in probability theory and its applications.