Probability Distributions Set 2 (Exponential Distribution)

The previous article covered the basics of Probability Distributions and talked about the Uniform Probability Distribution. This article covers the Exponential Probability Distribution which is also a Continuous distribution just like Uniform Distribution.

 

Introduction –

 

Suppose we are posed with the question- How much time do we need to wait before a given event occurs?
The answer to this question can be given in probabilistic terms if we model the given problem using the Exponential Distribution.
Since the time we need to wait is unknown, we can think of it 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.

 

 

Here  is the rate parameter and its effects on the density function are illustrated below –

 

 

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To check if the above function is a legitimate probability density function, we need to check if it’s integral over its support is 1.

 

 

 

Expected Value –

To find out the expected value, we simply multiply the probability distribution function with x and integrate over all possible values(support).

 

Variance and Standard deviation –

The variance of the Exponential distribution is given by-

 

 

 

 

Lack of Memory Property –

Now consider that in the above example, after detecting a particle at the 30 second mark, no particle is detected three minutes since.
Because we have been waiting for the past 3 minutes, we feel that a detection is due i.e. the probability of detection of a particle in the next 30 seconds should be higher than 0.3. However. this is not true for the exponential distribution. We can prove so by finding the probability of the above scenario, which can be expressed as a conditional probability-

 

 

The fact that we have waited three minutes without a detection does not change the probability of a detection in the next 30 seconds. Therefore, the probability only depends on the length of the interval being considered.