Exponential Distribution

The exponential distribution is a probability distribution which represents the time between events in a Poisson process. It is also called negative exponential distribution. It is a continuous probability distribution used to represent the time we need to wait before a given event happens. It is the constant counterpart of the geometric distribution, which is rather discrete. It also has the crucial property of being memoryless.

It is very much related to Poisson distribution. For example, if the number of deaths is modelled by Poisson distribution, then the time between each death is represented by an exponential distribution.

Exponential Distribution Graph

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 is 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=1 and λ=2.

Exponential Distribution Graph

Exponential Distribution Formula

There are a number of formulas defined for this distribution based on its characteristics. Let us learn them one by one.

Probability Density Function of an Exponential Distribution

The probability density function (pdf) of an exponential distribution is given by;

F(x;λ) = λe– λx when x ≥ 0,

F(x;λ) = 0 when x < 0.

Where ;

  • e is the natural number.
  • λ is the mean time between events and called a rate parameter. λ > 0
  • And x is any random variable called exponential random variable.

The exponential distribution shows infinite divisibility which is the probability distribution of the sum of an arbitrary number of independent and identically distributed random variables.

Cumulative Distribution Function of an Exponential Distribution

Similarly, the cumulative distribution function of an exponential distribution is given by;

F(x;λ) = 1-e– λx : x ≥ 0,

F(x;λ) = 0 : x < 0.

Exponential Distribution Mean or Expected Value

The expected value of an exponential random variable X with rate parameter λ is given by;

E[X] = 1/ λ

Exponential Distribution Variance

The variance of exponential random variable X is given by;

Var[X] = 1/λ2

Therefore, the standard deviation is equal to the mean.

Exponential Distribution Moment Generating Function

The ‘moment generating function’ of an exponential random variable X for any time interval t<λ, is defined by;

MX(t) = λ/λ-t

Memorylessness Property

This distribution has a memorylessness, which indicates it “forgets” what has occurred before it. We can say if we continue to wait, the length of time we wait for, neither increases nor decreases the likelihood of an event occurring. Any time may be considered as time zero. It is defined as;

P(X ≤ x+y|X>x) = P(X≤y)

X is the time we need to wait before a specific event happens. The above expression defines the possibility that the event occurred during a time interval of length ‘t’ is independent of how much time has already passed (x) without the event happening.

Sum of Exponential Random Variable

The sum of an exponential random variable or also called Gamma random variable of an exponential distribution having a rate parameter ‘λ’ is defined as;
sum of exponential distribution
Where Z is the gamma random variable which has parameters 2n and n/λ and Xi = X1X2, …, Xn are n mutually independent variables.

Exponential Distribution Applications

One of the widely used continuous distribution is exponential distribution. It helps to determine the time elapsed between the events. It is used in the range of applications such as reliability theory, queuing theory, physics and so on. Some of the fields that are modelled by the exponential distribution are as follows:

  • Exponential distribution helps to find the distance between mutations on a DNA strand
  • Calculating the time until the radioactive particle decays.
  • Helps on finding the height of different molecules in a gas at the stable temperature and pressure in a uniform gravitational field
  • Helps to compute the monthly and annual highest values of regular rainfall and river outflow volumes

Exponential Distribution Problems

Example:

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

Solution:

It is given that, 2 phone calls per hour. So, it would expect that one phone call at every half-an-hour. So, we can take

λ = 0.5

So, the computation is as follows:

\(p (0\leq X\leq 1) =\sum_{x=0}^{1}0.5e^{-0.5x}\)

= 0.393469

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

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