What is exponential distribution

In summary, the exponential distribution is a probability distribution that describes the likelihood of failure of a machine at any given time, with a probability density function of e^{-\lambda t} \lambda . It is commonly used to model random time intervals and can be used to calculate the probability of failure in a given time interval.
  • #1
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Definition/Summary

The exponential distribution is a probability distribution that describes a machine that it equally likely to fail at any given time.

Equations

[tex] f(t) = e^{-\lambda t} \lambda [/tex]

Extended explanation

A machine is equally likely to fail at any given time. For any [itex]t[/itex], the probability of failure in the interval [itex](t, t + dt)[/itex] is [itex]\lambda dt[/itex]. So the probability that it doesn't fail in that interval must be [itex]1 - \lambda dt[/itex]

Let us calculate the probability that it doesn't fail in the interval [itex](0,t)[/itex]. Divide [itex]t[/itex] into [itex]n[/itex] equal parts. Each part then has size [itex]\frac{t}{n}[/itex]. The probabilities that it doesn't fail in the intervals [itex](0,\frac{t}{n})[/itex], [itex](0,2\frac{t}{n})[/itex], ... are

[tex] 1 - \lambda \frac{t}{n} [/tex]
[tex] (1 - \lambda \frac{t}{n})^2 , [/tex]

... respectively. Therefore we find that the probability that it doesn't fail in the interval [itex](0,t)[/itex] is approximately

[tex] (1 - \lambda \frac{t}{n})^n .[/tex]

The exact answer is the limit of the above expression as [itex] n \rightarrow \infty[/itex], i.e.

[tex]e^{-\lambda t}.[/tex]

We can now use this to find the probability density function [itex]f(t)[/itex] that it fails for the first time in the interval [itex](t,t+dt)[/itex]. Clearly, this is the probability that it doesn't fail in the interval [itex](0,t)[/itex] times the probability that it fails in the interval [itex](t,t+dt)[/itex].

[tex]f(t) = e^{-\lambda t} \lambda.[/tex]

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  • #2
The exponential distribution is often used as a model of the duration of random time intervals, such as

  • Time between two calls
  • Waiting time in a line
  • Longevity of atoms in radioactive decay
  • Lifetime of components, machinery and equipment when aging phenomena do not need to be considered
  • as a rough model for small and medium damages in households
  • motor vehicle liability
##\lambda## represents the number of expected events per unit interval.
 

What is exponential distribution?

Exponential distribution is a probability distribution that describes the time between events in a process where the probability of an event occurring is constant. It is a continuous distribution and is often used to model and analyze the time between occurrences of rare events.

What are the characteristics of exponential distribution?

The exponential distribution is characterized by its rate parameter, which determines the shape and scale of the curve. It is a continuous distribution with a probability density function that is positive for all values of x and has a mean and standard deviation that are both equal to 1 divided by the rate parameter.

What is the difference between exponential and normal distributions?

The main difference between exponential and normal distributions is that the exponential distribution is a continuous distribution that is skewed to the right, while the normal distribution is symmetric and bell-shaped. Additionally, the exponential distribution is used to model the time between events, while the normal distribution is used to model a wide range of continuous data.

How is exponential distribution used in real life?

Exponential distribution is commonly used in fields such as economics, engineering, and biology to model the time between events. For example, it can be used to model the time between customer arrivals, the time between machine failures, or the time between radioactive decay events.

What is the relationship between exponential distribution and Poisson distribution?

The exponential distribution and Poisson distribution are closely related, as the Poisson distribution can be thought of as the discrete version of the exponential distribution. Both distributions are used to model the time between events, with the exponential distribution used for continuous data and the Poisson distribution used for discrete data.

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