The exponential distribution models the time until an event occurs, like product lifetimes or customer arrivals. It's defined by its probability density function and cumulative distribution function, with key parameters including the rate parameter and mathematical constant e.

This distribution has a unique memoryless property, making it ideal for modeling situations with constant event rates over time. It's closely related to the Poisson distribution, with applications in fields like survival analysis and reliability engineering.

The Exponential Distribution

Exponential distribution probability calculations

• Models time until an event occurs (product lifetimes, customer arrivals)
• Probability density function (PDF): $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
  • $\lambda$: rate parameter, average events per unit time
  • $e$: mathematical constant ≈ 2.71828
• Cumulative distribution function (CDF): $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
  • Probability event occurs before time $x$
• Probability of event in time interval $[a, b]$: $P(a \leq X \leq b) = F(b) - F(a) = e^{-\lambda a} - e^{-\lambda b}$
• Mean: $\frac{1}{\lambda}$, variance: $\frac{1}{\lambda^2}$
• Examples:
  • Time until a light bulb fails (product lifetime)
  • Time between customer arrivals at a store (interarrival time)

Memoryless property applications

• Unique characteristic of exponential distribution
• Probability of event in next time interval independent of time passed
• Mathematically: $P(X > s + t | X > s) = P(X > t)$ for non-negative $s$ and $t$
• Device longevity or service times applications:
  • Probability of device failing in next hour same regardless of functioning time
  • Probability of customer arriving in next 5 min same regardless of time since last arrival
• Suitable for modeling situations with constant event rate over time, independent of age or history
• Examples:
  • Lifespan of a car battery (device longevity)
  • Time between calls at a call center (service times)

Poisson vs exponential distributions

• Poisson distribution: discrete, models number of events in fixed time or space interval
• Exponential distribution: continuous, models time between events in Poisson process
• In Poisson process with rate $\lambda$:
  • Number of events in fixed time $t$ follows Poisson distribution with parameter $\lambda t$
  • Time between consecutive events follows exponential distribution with rate $\lambda$
• Converting event frequencies and time intervals:
  1. Average events per unit time $\lambda$ ⟷ Average time between events $\frac{1}{\lambda}$
  2. Average time between events $\mu$ ⟷ Average events per unit time $\frac{1}{\mu}$
• Models situations with interest in either:
  • Number of events in fixed time (Poisson)
  • Time between consecutive events (exponential)
• Examples:
  • Number of customers arriving at a store in an hour (Poisson)
  • Time between earthquakes in a region (exponential)

Additional Applications and Concepts

• The exponential distribution is a continuous probability distribution used in various fields
• Survival analysis: Models time until an event occurs, such as failure or death
  • Hazard function: Represents the instantaneous rate of occurrence of the event
• Exponential random variable: A random variable that follows the exponential distribution