Exponential Distribution

The Exponential Distribution is a continuous probability distribution that models the time or distance between independent events that occur at a constant average rate λ.

It is the continuous counterpart of the Poisson distribution.

In simple terms:

“If events happen randomly and independently at a steady rate,
the Exponential Distribution tells us how long we’ll wait until the next event.”

Probability Density Function (PDF)

Where:

• x = time or distance between events
• λ = rate parameter (average number of events per unit time)
• e = 2.718 (Euler’s number)

The total area under the curve = 1

Cumulative Distribution Function (CDF)

It represents the probability that the event occurs within time x.

Examples

• Bus arrivals - Average 4 buses/hour, mean waiting time 15 minutes
• Customer arrivals - 2 per minute, mean waiting time 0.5 minutes
• Machine failures - 1 failure every 10 hours, mean waiting time 10 hours
• Call centre - 5 calls per 10 minutes, mean waiting time 2 minutes

• Left plot (PDF):
• Starts high at x=0 and declines exponentially — most events happen soon after the last one.
• The curve never touches zero but approaches it as x increases.
• Right plot (CDF):
• Starts at 0 and rises steeply toward 1 — showing the probability of the event occurring by time x.

The faster the event rate (higher λ), the steeper the curve.
Slower rates (smaller λ) make the curve flatter — longer average wait times

Effect of λ (Rate Parameter)

Larger λ → shorter expected time (events occur more frequently)
Smaller λ → longer expected time (events are rarer)

In Machine Learning

• Modelling time-to-event data - Used in survival analysis or reliability modelling
• Queueing & network systems - Modelling inter-arrival or service times
• Simulation / Monte Carlo methods - Random waiting times between events
• Poisson process foundation - Time between Poisson events follows exponential distribution
• Hazard rate modelling - Used in probabilistic and time-based ML models