# Probability Distribution – Exponential

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149 Probability Distribution – Exponential: The Exponential Probability Distribution, models the time elapsed between events in a Poisson Process (Poisson distribution).

Let`s say, Poisson Distribution models the number of births in a given time period then exponential distribution models the time in between each birth. Where b is called base and x is the input variable that occurs as exponent.

Exponential Distribution: Since, Exponential Probability distribution is used to model time elapsed between two events, we can think of time elapsed as a random variable. If the probability of the event happening in a given interval is proportional to the length of the interval, then the Random Variable has an exponential distribution. The support (set of values the Random Variable can take) of an Exponential Random Variable is the set of all positive real numbers.

Rx = [0 , ∞)

Probability Density function: For a positive real number λ, the probability density function of a Exponentially distributed Random variable is given by- Here λ is the rate parameter and its effects on the density function are illustrated below: 