Exponential

The exponential distribution is a continuous probability distribution commonly used to model the time between events in a Poisson process. It is characterized by a single parameter $\lambda >0$, known as the rate parameter.

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Probability Density Function (PDF)

The probability density function of the exponential distribution is:

$$f_X(x)=\{\begin{array}{ll}\lambda e^{-\lambda x},& x\ge 0,\\ 0,& x<0.\end{array}$$

• $x$ represents the time or distance between events.
• $\lambda$ controls the rate of decay; higher $\lambda$ values result in a steeper decline.

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Cumulative Distribution Function (CDF)

The cumulative distribution function is the integral of the PDF:

$$F_X(x)={\int }_0^x\lambda e^{-\lambda t}dt=\{\begin{array}{ll}1-e^{-\lambda x},& x\ge 0,\\ 0,& x<0.\end{array}$$

The CDF represents the probability that $X\le x$.

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Expectation (Mean)

The expected value of an exponential random variable $X$ is given by:

$$E[X]=\frac{1}{\lambda }.$$

This is the average time or distance between events.

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Variance

The variance of an exponential random variable $X$ is:

$$\text{Var}(X)=\frac{1}{{\lambda }^2}$$

The standard deviation is the square root of the variance, ${\sigma }_X=\frac{1}{\lambda }$.

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Key Properties

1. Memorylessness: The exponential distribution is the only continuous distribution with the property that:

$$\mathrm{Pr}(X>x+t\mid X>t)=\mathrm{Pr}(X>x),\forall x,t\ge 0.$$

This means that the distribution does not “remember” past events.

2. Relationship with the Poisson Process: If the time between events follows an exponential distribution with rate $\lambda$, the number of events in a time interval of length $t$ follows a Poisson distribution with mean $\lambda t$.