Exponential Distribution definition, formula with applications

In probabilistic statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process where events occur continuously and independently at a constant average rate. It is a special case of the gamma distribution.

The exponential distribution is frequently used to calculate the amount of time until a certain event occurs. The amount of time (from now) until an earthquake occurs, for example, has an exponential distribution. Other examples include the length of long-distance business phone calls in minutes and the time a car battery lasts in months. It may also be demonstrated that the value of the change in your pocket or handbag follows a roughly exponential distribution.

What is the Exponential Distribution?

At its heart, the exponential distribution describes the time until a single event happens. Imagine you’re waiting for a bus at a bus stop. If the bus arrivals follow a Poisson process (buses arrive randomly and independently with a consistent average frequency), then the time you wait for the next bus is modeled by an exponential distribution.

A continuous random variable X is said to have an exponential distribution with parameter θ if its probability density function is defined as,

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

where, θ is the only parameter of the distribution and θ>0. Exponential distribution is continuous probability distribution and it has memoryless property like geometric distribution.

Characteristics of Exponential Distribution

There are some important characteristics as following,

Exponential distribution has only one parameter ‘λ’.
Mean of exponential distribution (variate) is 1/λ.
Variance of exponential distribution (variate) is 1/λ².
Moments of all order exists in exponential distribution.
Characteristic function of exponential distribution is $frac{lambda}{lambda-it}$ .
Moment generating function of exponential distribution is $frac{lambda}{lambda-t}, text{ for } t < lambda$ .
Median of exponential distribution is also 1/λ.
The measure of skewness β₁= 4.
Measure of kurtosis, β₂= 6.These measures show that exponential distribution is positive skewed and leptokurtic.
If the value of λ=1; mean= variance, if λ<1; mean<variance and if λ>1; mean>variance.
It also process the memoryless property just like geometric distribution.

Applications of the Exponential Distribution

The exponential distribution is widely used in various fields due to its ability to model waiting times and event durations. Here are some common applications:

Reliability Engineering: Modeling the time until a component fails. For example, predicting how long a lightbulb will last or the time until a hard drive crashes.
Queueing Theory: Analyzing waiting times in queues, such as customer service lines or call centers. Understanding average waiting times helps optimize resource allocation.
Telecommunications: Modeling the time between phone calls or data packets arriving at a server.
Finance: Assessing the time until a financial event occurs, such as a stock price reaching a certain threshold or a loan defaulting.
Medicine: Modeling the survival time of patients after a certain treatment.
Radioactive Decay: Describing the time until a radioactive atom decays.
Meteorology: Analyzing the time between major weather events like hurricanes or floods.

Examples and Calculations

Let’s illustrate the exponential distribution with some examples:

Example 1: Lightbulb Lifespan

Suppose a lightbulb has a lifespan that follows an exponential distribution with a rate parameter of λ = 0.001 failures per hour.

What is the average lifespan of the light bulb?

-E[X] = 1/λ = 1/0.001 = 1000 hours.

What is the probability that the lightbulb lasts longer than 1500 hours?

-We need to calculate P(X > 1500). This is equivalent to 1 – P(X ≤ 1500) = 1 – F(1500).P(X > 1500) = 1 – F(1500) = 1 – (1 – e^(-0.001 * 1500)) = e^(-0.001 * 1500) ≈ 0.223So, there’s approximately a 22.3% chance the lightbulb will last longer than 1500 hours.

What is the probability that the lightbulb fails within the first 500 hours?

-We need to calculate P(X ≤ 500) = F(500).F(500) = 1 – e^(-0.001 * 500) ≈ 0.393So, there’s approximately a 39.3% chance the lightbulb will fail within the first 500 hours.

Example 2: Call Center

A call center receives calls at an average rate of 5 calls per minute. What is the probability that the time until the next call is less than 10 seconds (1/6 of a minute)?

Here, λ = 5 calls/minute.
We want to find P(X < 1/6).F(1/6) = 1 – e^(-5 * (1/6)) ≈ 0.565. Therefore, the probability that the time until the next call is less than 10 seconds is approximately 56.5%.

Memorylessness of Exponential Distribution

Assume it’s been five minutes since the last customer arrived. Given the unusually long period of time that has passed, it appears that a customer will arrive within the next minute. This is not the case with the exponential distribution. The additional time spent waiting for the next customer is independent of how much time has passed since the previous customer. The memoryless property is what it’s called. The memoryless property, in particular, states that,

P (X > r + t | X > r) = P (X > t) for all r ≥ 0 and t ≥ 0

For example, if the previous client arrived five minutes ago. The chance that the next customer would arrive in more than one minute is estimated using r = 5 and t = 1 in the above equation.

P(X > 5 + 1 | X > 5) = P(X > 1) = e(–0.5)(1) ≈ 0.6065.

The memoryless property states that knowing about previous events has no bearing on future possibilities. In this example, it means that an old part is no more likely than a brand new part to break down at any given moment. To put it another way, the part remains as good as new until it breaks. If a part has already lasted ten years, the likelihood that it will last another seven is P(X > 17|X > 10) =P(X > 7) = 0.4966.

Relation between Poisson and Exponential distribution

The exponential distribution and the Poisson distribution have an interesting relationship. Assume that the amount of time that passes between two events follows an exponential distribution with a mean μ of units of time. Assume that these times are unrelated, that is, the time between events is unaffected by the time between preceding events. If these assumptions are correct, the number of occurrences per unit time will follow a Poisson distribution with a mean of 1/μ. Remember that if X has a Poisson distribution with mean, P(X=k)=[λ^k.e^(−λ)]/k! .

In contrast, if the number of occurrences per unit time follows a Poisson distribution, the time between events will follow an exponential distribution. (k!=k*(k-1*)) (k–2)*(k-3)… 3*2*1).

Suitable Scenarios

Events occur randomly and independently: The memoryless property relies on the assumption that past events don’t influence future events.
Constant average rate: The rate parameter λ must be relatively constant over time.
Modeling time until a single event: The exponential distribution describes the time until the first event, not the time until the nth event.

Inappropriate Scenarios

Events are not independent: If the occurrence of one event affects the probability of future events, the exponential distribution is not appropriate. For example, if a machine failure makes subsequent failures more likely.
Rate is not constant: If the rate of events changes significantly over time, you might need to consider a non-homogeneous Poisson process or a different distribution altogether.
Modeling the time until the nth event: The Gamma distribution is used for modeling the time until the nth event in a Poisson process.
Events have a minimum waiting time: If there’s a minimum time that must pass before an event can occur, the exponential distribution won’t fit the data well.

Conclusion

The exponential distribution is a versatile and essential tool for modeling waiting times and event durations. Understanding its properties, especially the memoryless property, and its relationship to the Poisson process is crucial for applying it correctly. While its assumptions of independence and constant rate must be carefully considered, the exponential distribution remains a fundamental building block in probability, statistics, and various applied fields. By mastering the concepts outlined in this article, you’ll be well-equipped to leverage the power of the exponential distribution in your own analyses and modeling endeavors. Data Science Blog