5.3 The Exponential Distribution

The exponential distribution is a crucial tool in business statistics, modeling the time until an event occurs or the time between events. It's particularly useful for analyzing customer arrivals, machine failures, and product lifetimes, providing insights into timing and probability.

One of the distribution's key features is its memoryless property, which means the probability of a future event is independent of time already passed. This makes it ideal for modeling systems with constant failure rates, like electronic components or customer service scenarios.

Properties and Applications of the Exponential Distribution

Exponential distribution probability calculations

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• Models time until an event occurs or time between events using continuous probability distribution
  • Exponential random variable $X$ represents waiting time until event happens
  • Probability density function (PDF): $f(x) = \lambda [e](https://www.fiveableKeyTerm:e)^{-\lambda x}$ for $x \geq 0$, where $\lambda > 0$ is rate parameter
  • Cumulative distribution function (CDF): $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
• Calculate probability of event occurring within specific time interval using CDF
  • $P(X \leq x) = F(x) = 1 - e^{-\lambda x}$
  • If average time between customer arrivals is 10 minutes ($\lambda = \frac{1}{10}$), probability next customer arrives within 5 minutes is $P(X \leq 5) = 1 - e^{-\frac{1}{10} \cdot 5} \approx 0.3935$
• Calculate probability of event occurring after specific time using complement of CDF
  • $P(X > x) = 1 - F(x) = e^{-\lambda x}$
  • Using same scenario, probability next customer arrives after 15 minutes is $P(X > 15) = e^{-\frac{1}{10} \cdot 15} \approx 0.2231$
• The time between successive events in a Poisson process is called the interarrival time and follows an exponential distribution

Memoryless property of exponential distribution

• Exponential distribution has memoryless property, meaning probability of future event is independent of time already passed
  • Mathematically, $P(X > s + t | X > s) = P(X > t)$ for all $s, t \geq 0$
  • Probability of waiting additional time $t$ given you've already waited time $s$ is same as probability of waiting time $t$ from start
• Memoryless property suitable for modeling longevity of devices or systems with constant failure rate
  • If light bulb has exponential lifetime distribution with average lifespan of 1000 hours ($\lambda = \frac{1}{1000}$), probability it lasts additional 500 hours given it's been in use for 200 hours is same as probability of new light bulb lasting 500 hours
    • $P(X > 700 | X > 200) = P(X > 500) = e^{-\frac{1}{1000} \cdot 500} \approx 0.6065$
• Memoryless property also applies to time between events in Poisson process (customer arrivals, machine failures)

Exponential vs Poisson distributions

• Exponential and Poisson distributions closely related, both describe occurrence of events over time
• Exponential distribution (continuous):
  • Models time until event occurs or time between events
  • PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$, where $\lambda > 0$ is rate parameter
  • CDF: $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
  • Mean: $E(X) = \frac{1}{\lambda}$
  • Variance: $Var(X) = \frac{1}{\lambda^2}$
• Poisson distribution (discrete):
  • Models number of events occurring in fixed interval of time or space
  • Probability mass function (PMF): $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$ for $k = 0, 1, 2, \ldots$, where $\lambda > 0$ is average number of events per interval
  • Mean: $E(X) = \lambda$
  • Variance: $Var(X) = \lambda$
• Relationship between exponential and Poisson distributions:
  • If time between events follows exponential distribution with rate $\lambda$, then number of events in fixed time interval follows Poisson distribution with mean $\lambda t$, where $t$ is length of time interval
  • If customer arrivals follow Poisson process with average of 6 customers per hour ($\lambda = 6$), then time between customer arrivals follows exponential distribution with rate $\lambda = 6$ per hour ($\frac{1}{10}$ per minute)
• Applications:
  • Exponential distribution: Modeling lifetime of devices, waiting times between events, time until failure in reliability analysis
  • Poisson distribution: Modeling number of rare events in fixed interval (defects in product batch, customers arriving at store, accidents at intersection)

Additional Applications and Related Concepts

• Survival analysis: Uses exponential distribution to model time until an event of interest (e.g., failure, death) occurs
• Hazard function: Represents the instantaneous rate of failure at a given time, which is constant for the exponential distribution
• Exponential decay: Describes processes where a quantity decreases at a rate proportional to its current value, following an exponential distribution pattern