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 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
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
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. |
Mean or Expected Value
The expected value of an exponential random variable X with rate parameter λ is given by;
E[X] = 1/ λ |
Variance
The variance of exponential random variable X is given by;
Var[X] = 1/λ^{2} |
Therefore, the standard deviation is equal to the mean.
Moment Generating Function
The ‘moment generating function’ of an exponential random variable X for any time interval t<λ, is defined by;
M_{X}(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;
Where Z is the gamma random variable which has parameters 2n and n/λ and X_{i} = X_{1},X_{2},…,X_{n} are n mutually independent variables.