What Is Exponential Distribution?

(1) What Is Exponential Distribution?

The Exponential Distribution is a continuous distribution that is commonly used to measure the expected time for an event to occur.
For example, in physics, it is often used to measure radioactive decay.
In engineering, it is used to measure the time associated with receiving a defective part on an assembly line.
In finance, it is often used to measure the likelihood of the next default for a portfolio of financial assets.
For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution.
Other examples include the length, in minutes, of long-distance business telephone calls, and the amount of time, in months, a car battery lasts.

where e represents a natural number
λ = mean time between the events, also known as the rate parameter, and is λ > 0
x = random variable

The continuous random variable, say X is said to have an exponential distribution if it has the above probability density function.

The exponential distribution graph is a graph of the probability density function which shows the distribution of distance or time taken between events.
The two terms used in the exponential distribution graph are lambda (λ)and x.
Here, lambda represents the events per unit time and x represents the time.
The following graph shows the values for λ=0.5, λ=1.0, and λ=1.5.

Assume that, you usually get 2 phone calls per hour. calculate the probability, that a phone call will come within the next hour.

It is given that, 2 phone calls per hour. So, it would expect that one phone call every half an hour.
λ = 0.5
So, the computation is as follows:

Therefore, the probability of arriving the phone calls within the next hour is 0.393469

Let us determine the amount of time taken (in minutes) by office personnel to deliver a file from the manager’s desk to the clerk’s desk.
The function of time taken is assumed to have an exponential distribution with the average amount of time equal to 5 minutes.
Also, x is a continuous random variable. Based on the given data, determine the exponential distribution.

Given:
x = time taken to deliver a file in minutes
μ = 5 minutes
Therefore, the scale parameter is:
λ = 1 / μ
λ = 1 / 5 = 0.20
Hence, the exponential distribution probability function can be derived as,
f (x; λ) = 0.20 e ^{– 0.20*x}
Now, calculate the probability function at different values of x to derive the distribution curve.
For x = 0
For x = 0, f(0) = 0.20 e ^-0.20*0 = 0.200
For x = 1, f(1) = 0.20 e ^-0.20*1 = 0.164
For x = 2, f(2) = 0.20 e ^-0.20*2 = 0.134

All these questions concern the time we need to wait before a given event occurs.

If this waiting time is unknown, it is often appropriate to think of it as a random variable having an exponential distribution.