Instead of a probability mass function, a continuous random variable has a probability density function. This might seem like unnecessary nomenclature, but the density function and the mass function are very different creatures. An example of continuous random variable is a random variable with exponential density. The density function for an exponential random variable looks like this:

Like a Poisson random variable, an exponential random variable can take on only non-negative values. But unlike a Poisson variable, the exponential can take on any non-negative values, including non-integral values such as 4.25 or 5.612401. This property makes it a poor choice for count data, which must be an integer, but a great choice for time data, temperature data (measured in Kelvins, of course), or any other precise and positive variable.

When a random variable Z has a exponential distribution, we say it’s exponential and we denote it with $Z \~ Exp(\lambda)$

Its expected value is $E[Z|\lambda] = \frac{1}{\lambda}$

References

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