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Exponential Distribution Formula

The exponential distribution in probability is the probability distribution that describes the time between events in a Poisson process.

Probability Density Function

\[\large f(x ; \lambda ) = \left\{\begin{matrix} \lambda e^{-\lambda x} & x\geqslant 0,\\ 0 & x < 0. \end{matrix}\right.\]

Cumulative Distribution Function

\[\large F(x ; \lambda ) = \left\{\begin{matrix} 1 – e^{-\lambda x} & x\geqslant 0,\\ 0 & x < 0. \end{matrix}\right.\]

where $\lambda > 0$ is called the rate of the distribution.

The mean of the Exponential ($\lambda$) Distribution is calculated using integration by parts as –

\[\large E(X) = \int_{0}^{\infty } x\lambda e^{-\lambda x} \; dx\]

\[\large = \lambda \left [ \frac{-x \; e^{-\lambda x}}{\lambda}|_{0}^{\infty } + \frac{1}{\lambda }\int_{0}^{\infty } e^{-\lambda x} dx \right ]\]

\[\large = \lambda \left [ 0 + \frac{1}{\lambda }\frac{-e^{-\lambda x}}{\lambda} |_{o}^{\infty }\right ]\]

\[\large = \lambda \frac{1}{\lambda ^{2} }\]

\[\large = \frac{1}{\lambda }\]

 

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