Editing Exponential distribution (section)

===Sum of two independent exponential random variables===
The probability distribution function (PDF) of a sum of two independent random variables is the [[convolution of probability distributions|convolution of their individual PDFs]].  If <math>X_1</math> and <math>X_2</math> are independent exponential random variables with respective rate parameters <math>\lambda_1</math> and <math>\lambda_2,</math> then the probability density of <math>Z=X_1+X_2</math> is given by
<math display="block"> \begin{align}
 f_Z(z) &= \int_{-\infty}^\infty f_{X_1}(x_1) f_{X_2}(z - x_1)\,dx_1\\
   &= \int_0^z \lambda_1 e^{-\lambda_1 x_1} \lambda_2 e^{-\lambda_2(z - x_1)} \, dx_1 \\
   &= \lambda_1 \lambda_2 e^{-\lambda_2 z} \int_0^z e^{(\lambda_2 - \lambda_1)x_1}\,dx_1 \\
   &= \begin{cases}
        \dfrac{\lambda_1 \lambda_2}{\lambda_2-\lambda_1} \left(e^{-\lambda_1 z} - e^{-\lambda_2 z}\right) & \text{ if } \lambda_1 \neq \lambda_2 \\[4 pt]
        \lambda^2 z e^{-\lambda z} & \text{ if } \lambda_1 = \lambda_2 = \lambda.
      \end{cases}
 \end{align} </math>
The entropy of this distribution is available in closed form: assuming <math>\lambda_1 > \lambda_2</math> (without loss of generality), then
<math display="block">\begin{align}
 H(Z) &= 1 + \gamma + \ln \left( \frac{\lambda_1 - \lambda_2}{\lambda_1 \lambda_2} \right) + \psi \left( \frac{\lambda_1}{\lambda_1 - \lambda_2} \right) ,
\end{align}</math>
where <math>\gamma</math> is the [[Euler-Mascheroni constant]], and <math>\psi(\cdot)</math> is the [[digamma function]].<ref>{{cite arXiv|last1=Eckford |first1=Andrew W. |last2=Thomas |first2=Peter J. |date=2016 |title=Entropy of the sum of two independent, non-identically-distributed exponential random variables |class=cs.IT |eprint=1609.02911}}</ref>

In the case of equal rate parameters, the result is an [[Erlang distribution]] with shape 2 and parameter <math>\lambda,</math> which in turn is a special case of [[gamma distribution]].

The sum of n independent Exp(''λ)'' exponential random variables is Gamma(n, ''λ)'' distributed.