17.6 Examples of Exponential Distribution

The Exponential Distribution is a continuous probability distribution that models the time between events in a Poisson process. It's particularly useful for situations where events occur at a constant average rate. Common applications include reliability engineering, analyzing waiting times in queuing systems, and predicting service times.

Core Concepts

Probability Density Function (PDF)

The PDF describes the likelihood of a specific value occurring. For the exponential distribution, it's given by:

$f(x; \lambda) = \lambda e^{-\lambda x}$, for $x \geq 0$

Where:

• $\lambda$ (lambda) is the rate parameter. It represents the average number of events per unit of time.
• $x$ is the time or interval between events.

Cumulative Distribution Function (CDF)

The CDF gives the probability that the random variable (time) is less than or equal to a specific value.

$F(x) = P(X \leq x) = \int_{0}^{x} \lambda e^{-\lambda t} dt = 1 - e^{-\lambda x}$

Survival Function (or Tail Probability)

The survival function, $P(X > x)$, gives the probability that the time between events is greater than a specific value.

$P(X > x) = 1 - F(x) = 1 - (1 - e^{-\lambda x}) = e^{-\lambda x}$

This is often used to calculate the probability of an event occurring after a certain time.

Examples

Example 1: ATM Waiting Time

Problem: The waiting time at a bank's ATM follows an exponential distribution with a rate parameter $\lambda = 5$ (events per minute). What is the probability that a customer completes their transaction in less than 3 minutes?

Solution:

• Given:
  • Rate parameter $\lambda = 5$ per minute.
  • Time of interest $x = 3$ minutes.
• Find: $P(X < 3)$

We use the CDF to find the probability of the time being less than a specific value: $P(X < 3) = F(3) = 1 - e^{-\lambda x}$ $P(X < 3) = 1 - e^{-(5 \times 3)}$ $P(X < 3) = 1 - e^{-15}$

Calculating the value: $P(X < 3) \approx 1 - 0.000000305 \approx 0.9999997$

Answer: The probability that the transaction takes less than 3 minutes is approximately 0.9999997, or very close to 0.99.

Example 2: Call Arrival Time at Call Centre

Problem: Calls to a life insurance company's helpdesk occur at an average rate of $\lambda = 4$ per hour. What is the probability that the next call arrives after more than 2 hours?

Solution:

• Given:
  • Rate parameter $\lambda = 4$ per hour.
  • Time of interest $x = 2$ hours.
• Find: $P(X > 2)$

We use the survival function to find the probability of the time being greater than a specific value: $P(X > 2) = e^{-\lambda x}$ $P(X > 2) = e^{-(4 \times 2)}$ $P(X > 2) = e^{-8}$

Calculating the value: $P(X > 2) \approx 0.000335$

Answer: The probability that the next call arrives after more than 2 hours is approximately 0.000335.

Summary of Key Formulas

• Probability Density Function (PDF): $f(x) = \lambda e^{-\lambda x}$, for $x \geq 0$

• Cumulative Distribution Function (CDF): $F(x) = P(X \leq x) = 1 - e^{-\lambda x}$

• Survival Function (Tail Probability): $P(X > x) = e^{-\lambda x}$

Important Properties

• Memorylessness: The exponential distribution is famously "memoryless." This means that the probability of an event happening in the future is independent of how much time has already passed. For example, if you've already waited 10 minutes for an ATM transaction (and it follows an exponential distribution), the probability of it finishing in the next minute is the same as the probability of a new customer finishing their transaction within the first minute. This is a key characteristic often discussed in reliability and queuing theory.

• Relationship to Poisson Process: If the number of events occurring in a fixed interval of time follows a Poisson distribution with rate $\lambda$, then the time between these events follows an exponential distribution with the same rate parameter $\lambda$.

Potential Interview Questions

• What is the Exponential distribution and when is it used? It's a continuous probability distribution modeling the time between events in a Poisson process, used for things like waiting times or component lifetimes.

• How is the Exponential distribution related to the Poisson process? If events occur according to a Poisson process with rate $\lambda$, the time intervals between those events are exponentially distributed with parameter $\lambda$.

• Write the PDF and CDF of the Exponential distribution and explain their components. (As detailed in the "Core Concepts" section above, explaining $\lambda$ as the rate and $x$ as time.)

• How do you calculate the probability that an event occurs within a certain time using the Exponential distribution? By using the CDF: $P(X \leq x) = 1 - e^{-\lambda x}$.

• What is the survival function in the Exponential distribution and when is it used? $P(X > x) = e^{-\lambda x}$. It's used to find the probability that the time between events exceeds a certain value.

• In what scenarios would the Exponential distribution be preferred over other distributions? When modeling the time between events that occur at a constant average rate, and when the "memoryless" property is a reasonable assumption (e.g., simple electronic component failures, inter-arrival times of customers to a system under stable conditions).

• Explain how the rate parameter $\lambda$ affects the shape of the Exponential distribution. A higher $\lambda$ means events occur more frequently, leading to a PDF that decays more rapidly and a distribution shifted towards shorter times. A lower $\lambda$ means events are less frequent, resulting in a slower decay and a distribution spread over longer times.

• What is memorylessness in the Exponential distribution, and why is it important? It means the past history of waiting time doesn't affect future probabilities. It simplifies calculations significantly but requires careful consideration of whether the real-world phenomenon truly exhibits this property.

• How would you apply the Exponential distribution in queueing theory or service systems? To model the time between customer arrivals or the time it takes to serve a customer, assuming these times are independent and occur at a constant average rate.

• Give a real-world example of using the Exponential distribution and solve it using the CDF or survival function. (As demonstrated in Example 1 and Example 2 provided above.)