1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The general formula for the probability
density function of the exponential distribution is
\( f(x) = \frac{1} {\beta} e^{(x  \mu)/\beta} \hspace{.3in} x \ge \mu; \beta > 0 \) where μ is the location parameter and β is the scale parameter (the scale parameter is often referred to as λ which equals 1/β). The case where μ = 0 and β = 1 is called the standard exponential distribution. The equation for the standard exponential distribution is \( f(x) = e^{x} \;\;\;\;\;\;\; \mbox{for} \; x \ge 0 \) The general form of probability functions can be expressed in terms of the standard distribution. Subsequent formulas in this section are given for the 1parameter (i.e., with scale parameter) form of the function. The following is the plot of the exponential probability density function.


Cumulative Distribution Function 
The formula for the cumulative distribution
function of the exponential distribution is
\( F(x) = 1  e^{x/\beta} \hspace{.3in} x \ge 0; \beta > 0 \) The following is the plot of the exponential cumulative distribution function.


Percent Point Function 
The formula for the percent point
function of the exponential distribution is
\( G(p) = \beta\ln(1  p) \hspace{.3in} 0 \le p < 1; \beta > 0 \) The following is the plot of the exponential percent point function.


Hazard Function 
The formula for the hazard
function of the exponential distribution is
\( h(x) = \frac{1} {\beta} \hspace{.3in} x \ge 0; \beta > 0 \) The following is the plot of the exponential hazard function.


Cumulative Hazard Function 
The formula for the cumulative hazard
function of the exponential distribution is
\( H(x) = \frac{x} {\beta} \hspace{.3in} x \ge 0; \beta > 0 \) The following is the plot of the exponential cumulative hazard function.


Survival Function 
The formula for the survival
function of the exponential distribution is
\( S(x) = e^{x/\beta} \hspace{.3in} x \ge 0; \beta > 0 \) The following is the plot of the exponential survival function.


Inverse Survival Function 
The formula for the inverse survival
function of the exponential distribution is
\( Z(p) = \beta\ln(p) \hspace{.3in} 0 \le p < 1; \beta > 0 \) The following is the plot of the exponential inverse survival function.


Common Statistics 


Parameter Estimation  For the full sample case, the maximum likelihood estimator of the scale parameter is the sample mean. Maximum likelihood estimation for the exponential distribution is discussed in the chapter on reliability (Chapter 8). It is also discussed in chapter 19 of Johnson, Kotz, and Balakrishnan.  
Comments  The exponential distribution is primarily used in reliability applications. The exponential distribution is used to model data with a constant failure rate (indicated by the hazard plot which is simply equal to a constant).  
Software  Most general purpose statistical software programs support at least some of the probability functions for the exponential distribution. 