Exponential Distribution Last Updated : 24 Jul, 2025 Comments Improve Suggest changes Like Article Like Report The Exponential Distribution is one of the most commonly used probability distributions in statistics and data science. It is widely used to model the time or space between events in a Poisson process. In simple terms, it describes how long you have to wait before something happens, like a bus arriving or a customer calling a help center.For example, if buses arrive at a bus stop every 15 minutes on average, the time you wait for the next bus can be modeled using an exponential distribution.Probability Density Function (PDF)The probability density function of the exponential distribution is:f(x; \lambda) = \lambda e^{-\lambda x}, \quad x \geq 0Where:\lambda > 0 is the rate parameter (how often events occur)x is the time or distance until the next eventCumulative Distribution Function (CDF)The cumulative distribution function gives the probability that the event occurs within time :F(x; \lambda) = 1 - e^{-\lambda x}, \quad x \geq 0Mean and VarianceMean (Expected Value): E[X] = \frac{1}{\lambda} Variance: \text{Var}(X) = \frac{1}{\lambda^2} Memoryless PropertyThe exponential distribution is memoryless, which means:P(X > s + t \mid X > s) = P(X > t)This property tells us that the probability of waiting longer does not depend on how long you've already waited. This is unique to the exponential distribution.ExampleSuppose calls come into a customer support center at an average rate of 2 per minute. What is the probability that you wait more than 30 seconds for the next call?Solution:Understand the Rate (λ): Since 2 calls come in per minute, that means the average rate is: λ=2 calls per minuteConvert Time: Find the probability of waiting more than 30 seconds. But since the rate is in minutes, convert 30 seconds to minutes: 30 seconds=0.5 minutesUse the Exponential Distribution formula:P(X > 0.5) = e^{-\lambda x} = e^{-2 \cdot 0.5} = e^{-1} \approx 0.3679 So, there is about a 36.79% chance that the next call comes after 30 seconds.Example in PythonBefore its implementation we should have some basic knowledge about numpy, matplotlib and seaborn. Python import numpy as np import matplotlib.pyplot as plt import seaborn as sns # Set the rate parameter lambda lam = 1.5 # Generate exponential data data = np.random.exponential(1/lam, 1000) # Plot the KDE and histogram sns.histplot(data, kde=True, stat="density") plt.title("Exponential Distribution (lambda=1.5)") plt.xlabel("Time") plt.ylabel("Density") plt.show() Output:Exponential DistributionNote: The exponential distribution is closely related to the Poisson distribution. If events occur following a Poisson process with rate , then the time between events is exponentially distributed with parameter .Real-World ApplicationsCall Centers: Time between incoming callsBank Queues: Time between customer arrivalsWeb Servers: Time between requestsManufacturing: Time until a machine part failsTransport: Time between buses or trains Comment More infoAdvertise with us C cmkmanwani Follow Improve Article Tags : Engineering Mathematics AI-ML-DS With Python Data Science Similar Reads Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. 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