Exponential Distribution Help

What do these properties mean?

• Fully Defined by its Mean

  This means that only one parameter is necessary to know in order to define the entire distribution - the mean. This means that if you know the mean of an exponential distribution, you know the entire distribution.

• Continuous Analog

  The exponential distribution is very similar to the discrete geometric distribution. However, the exponential distribution is measured continuously (meaning that all fractional numbers are included in the distribution). This is why it is referred to as the "continuous analog" of the geometric distribution - they are very similar distributions other than the way they are measured.

• Random Variable

  In the case of a continuous distribution, the random variable is defined as having the density of the distribution. So, in this case, since the value that you are interested in observing is the random variable, it has the density of the exponential distribution.

Example

When dealing with a continuous distribution, if you see or hear the words "during" or "every", you are probably dealing with an exponential distribution. This is because exponential distributions define the interval of a Poisson process, meaning that they define the interval over which discrete events are observed.

For example, imagine that a you are observing how often you receive a spam email. You know from previous observation that you expect to receive a spam email every 1.2 days. This would be an exponential distribution with 𝛽 = 1/1.2 or about 0.8333 (because that is the mean value of how often you receive a spam email). The only thing left would be to determine the length of time you would like to observe from the distribution, and that would be x (because it is the value that you are finding the probability density of). Note that you will not always receive a spam email in that amount of time - there is only some probability that you will.

Requirements

• The mean value of the exponential distribution must be greater than zero (𝛽 > 0).
• The value that you are interested in observing from the distribution must be greater than zero (x > 0).