Template:Exponential Probability Density Function

The Two-Parameter Exponential Distribution
The two-parameter exponential pdf is given by:


 * $$f(T)=\lambda {{e}^{-\lambda (T-\gamma )}},f(T)\ge 0,\lambda >0,T\ge 0\text{ or }\gamma $$

where $$\gamma $$ is the location parameter. Some of the characteristics of the two-parameter exponential distribution are [19]:
 * 1) The location parameter, $$\gamma $$, if positive, shifts the beginning of the distribution by a distance of $$\gamma $$ to the right of the origin, signifying that the chance failures start to occur only after $$\gamma $$ hours of operation, and cannot occur before.
 * 2) The scale parameter is $$\tfrac{1}{\lambda }=\bar{T}-\gamma =m-\gamma $$.
 * 3) The exponential $$pdf$$ has no shape parameter, as it has only one shape.
 * 4) The distribution starts at $$T=\gamma $$ at the level of $$f(T=\gamma )=\lambda $$ and decreases thereafter exponentially and monotonically as $$T$$ increases beyond $$\gamma $$ and is convex.
 * 5) As $$T\to \infty $$, $$f(T)\to 0$$.

The One-Parameter Exponential Distribution
The one-parameter exponential $$pdf$$ is obtained by setting $$\gamma =0$$, and is given by:


 * $$ \begin{align}f(T)= & \lambda {{e}^{-\lambda T}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}T}},

& T\ge 0, \lambda >0,m>0 \end{align} $$


 * where:

This distribution requires the knowledge of only one parameter, $$\lambda $$, for its application. Some of the characteristics of the one-parameter exponential distribution are [19]:
 * The location parameter, $$\gamma $$, is zero.
 * The scale parameter is $$\tfrac{1}{\lambda }=m$$.
 * As $$\lambda $$ is decreased in value, the distribution is stretched out to the right, and as $$\lambda $$ is increased, the distribution is pushed toward the origin.
 * This distribution has no shape parameter as it has only one shape, i.e. the exponential, and the only parameter it has is the failure rate, $$\lambda $$.
 * The distribution starts at $$T=0$$ at the level of $$f(T=0)=\lambda $$ and decreases thereafter exponentially and monotonically as $$T$$ increases, and is convex.
 * As $$T\to \infty $$, $$f(T)\to 0$$.
 * The $$pdf$$ can be thought of as a special case of the Weibull $$pdf$$ with $$\gamma =0$$ and $$\beta =1$$.