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

#### Probability Density Function

\(f(x; \lambda )=\left\{\begin{matrix} \lambda e^{-\lambda x} & x\geq 0\\ 0 & x<0 \end{matrix}\right.\)

#### Cumulative Distribution Function

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

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

The mean of theÂ **Exponential ($\lambda$) Distribution**Â is calculated using integration by parts as â€“

\(\begin{aligned} E[X] &=\int_{-\infty}^{\infty} x f_{X}(x) d x \\ &=\lambda \int_{0}^{\infty} x e^{-\lambda x} d x \\ &=\lambda\left\{\left[x \int e^{-\lambda x} d x\right]_{0}^{\infty}-\left[\int \frac{d}{d x} x\left(\int e^{-\lambda x} d x\right) d x\right]_{0}^{\infty}\right.\end{aligned}\)

Simplifying further,

\(\begin{array}{l}=\lambda\left\{\left[x \frac{e^{-\lambda x}}{-\lambda}\right]_{0}^{\infty}-\left[\int \frac{e^{-\lambda x}}{-\lambda} d x\right]_{0}^{\infty}\right\} \\ =\left[\frac{-x}{e^{\lambda x}}\right]_{0}^{\infty}+\left[\int e^{-\lambda x} d x\right]_{0}^{\infty} \\ =\left[\frac{-x}{e^{\lambda x}}\right]_{0}^{\infty}+\left[\frac{e^{-\lambda x}}{-\lambda}\right]_{0}^{\infty} \\ =[0-0]-\frac{1}{\lambda}[0-1] \\ =\frac{1}{\lambda}\end{array}\)