Mon. Dec 4th, 2023
The Exponential Distribution: Applications, Derivation and its understanding

We’re gonna see what exponential distribution is.First, you should know what it is and why it’s been applied. However, it actually sticks with you and you ’ll be a lot more likely to apply it in your own line of work, If you understand why.  I’m using it to represent it more mathematically.

What is exponential distribution?

The Exponential Distribution is a statistical probability distribution that describes the time between events in a Poisson point process, where events occur continuously and independently at a constant average rate. It is often used to model the lifetime of products or systems, the time between arrivals of customers or requests, and other similar phenomena. 

The distribution has a single parameter, which is the rate parameter that determines the average time between events. The Exponential Distribution is a continuous distribution that takes on values greater than or equal to zero.

What is Poisson Distribution?

  • The Poisson Distribution is a statistical probability distribution that is used to model the number of occurrences of an event in a fixed interval of time or space, given that the events occur randomly and independently of each other. 
  • The Poisson Distribution has a single parameter, λ (lambda), which represents the average rate of occurrence of the event. The distribution takes on integer values starting from zero, as the event can occur zero or more times in the interval. The probability of observing a specific number of events is given by the Poisson probability mass function.
  • The Poisson Distribution has many applications, including in queuing theory, reliability engineering, epidemiology, and finance, among others. It is often used to model rare events or phenomena,

Why did we’ve to construct Exponential Distribution? 

 To  prognosticate the  quantum of  staying time until the coming event( i.e., success, failure,  appearance,etc.).  

For  Example, we want to  predict the following: The  quantum of time until the  client finishes browsing and actually purchases a commodity in your store( success).  The  quantum of time until the  tackle on AWS EC2 fails( failure).  

The  quantum of time you need to  stay until the  machine arrives(  appearance).  also, my coming question is this Why is λ * e( − λt) the PDF of the time until the coming event happens?  And the follow- up question would be What does X  Exp( 025) mean?  Does the parameter0.25 mean0.25  twinkles, hours, or days, or is It 0.25 events?  From this point on, I ’ll assume you know Poisson distribution outside and out.However, this composition will give you a clear idea, If you do n’t.

One thing that would save you from the confusion  later about X  Exp( 025) is to flash back  that 0.25 isn’t a time duration, but it’s an event rate. For  illustration, your blog has 500 callers a day. That’s a rate. 

The number of  guests arriving at the store in an hour, the number of earthquakes per time, the number of auto accidents in a week, the number of typos on a  runner, the number of hairs  set up in Chipotle,etc., are all rates( λ) of the unit of time, which is the parameter of the Poisson distribution. 

Still, when we model the  ceased time between events, we tend to speak in terms of time  rather of rate,e.gThe number of times a computer can power on without failure is 10 times(  rather of saying 0.1 failure/ time, which is a rate), a  client arrives every 10  twinkles, major hurricanes come every 7 times, etc. 

When you see the  language — “ mean ” of the exponential distribution — 1/ λ is what it means.  The confusion starts when you see the term “ decay parameter ”, or 

indeed worse, the term “ decay rate ”, which is  constantly used in exponential distribution. 

The decay parameter is expressed in terms of time(e.g., every 10 mins, every 7 times,etc.), which is a complementary( 1/ λ) of the rate( λ) in Poisson. suppose about it If you get 3  guests per hour, it means you get one  client every1/3 hour.  

So, now you can answer the following: What does it mean for “ X  Exp( 025) ”?  It means the Poisson rate will be 0.25.

 During a unit time( either it’s a  nanosecond, hour or time), the event occurs 0.25 times on average. Converting this into time terms, it takes 4 hours( a complementary of0.25) until the event occurs, assuming your unit time is an hour. 

Where is it used?

Values for an exponential arbitrary variable have further small values and smaller large values. The machine that you’re staying for will presumably come within the coming 10 twinkles rather than the coming 60 twinkles. 

The machine comes in every 15 twinkles on average.( Assume that the time that elapses from one machine to the coming has exponential distribution, which means the total number of motorcars to arrive during an hour has Poisson distribution.) And I just missed the machine! The motorist was unkind. The moment I arrived, the motorist closed the door and left.

However, I’ve to call Uber or otherwise I ’ll be late. If the coming machine does n’t arrive within the coming ten twinkles. What’s the probability that it takes lower than ten nanoseconds for the coming machine to arrive?

Ninety percent of the motorcars arrive within how many twinkles of the former machine? 

 How long does it take for two motorcars to arrive? * Post your answers in the comment, if you want to see if your answer is correct. b) trustability( failure) modeling Since we can model the successful event( the appearance of the machine), why not the failure modeling — the quantum of time a product lasts? The number of hours that AWS tackle can run before it needs a renewal is exponentially distributed with a normal of 1,000 hours( about a time). 

1. You do n’t have a backup garçon and you need an continued,000- hour run. What’s the probability that you’ll be suitable to complete the run without having to renew the garçon? 

2. What’s the probability that the garçon does n’t bear a renew between 12 months and 18 months? Note that occasionally, the exponential distribution might not be applicable — when the failure rate changes throughout the continuance.

Still, it’ll be the only distribution that has this unique property– constant hazard rate.  Service time modeling( Queuing proposition) The service times of agents(e.g., how long it takes for a Chipotle hand to make me a burrito) can also be modeled as exponentially distributed variables. The total length of a process — a sequence of several independent tasks — follows the Erlang distribution the distribution of the sum of several independent exponentially distributed variables.


Well, The Exponential Distribution is a fundamental probability distribution used to model the time between events in a Poisson process.

It has a simple and elegant mathematical derivation, and it is characterized by a single parameter that represents the average rate of events. The Exponential Distribution has a wide range of applications in various fields, including engineering, finance, and computer science, among others.

Its intuitive understanding is crucial to interpreting its results accurately and making informed decisions based on them. The Exponential Distribution plays a critical role in the development and implementation of models and systems that involve time-based events, making it an essential tool for researchers, practitioners, and decision-makers alike.

By Hari Haran

I'm Aspiring data scientist who want to know about more AI. I'm very keen in learning many sources in AI.

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