Exponential Distribution definition, formula with applications

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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.

Definition

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

According to wikipedia, there are some applications as following,

It occurs naturally when describing the lengths of the inter-arrival times in a homogeneous Poisson process.
The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.
In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives.

Similar caveats apply to the following examples which yield approximately exponentially distributed variables:

The time until a radioactive particle decays, or the time between clicks of a Geiger counter
The time it takes before your next telephone call
The time until default (on payment to company debt holders) in reduced form credit risk modeling

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,

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).