- Hyper-exponential distribution
In
probability theory , a hyper-exponential distribution is acontinuous distribution such that theprobability density function of therandom variable X is given by:f_X(x) = sum_{i=1}^n f_{Y_i}(y) p_i,
where Y_i is an exponentially distributed random variable with rate parameter lambda,_i, and p_i is the probability that "X" will take on the form of the exponential distribution with rate lambda,_i. It is named the "hyper"-exponential distribution since its
coefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and thehypoexponential distribution , which has a coefficient of variation less than one. While theexponential distribution is the continuous analogue of thegeometric distribution , the hyper-exponential distribution is not analogous to thehypergeometric distribution . the hyper-exponential distribution is an example of amixture density .An example of a hyper-exponential random variable can be seen in the context of
telephony , where, if someone has a modem and a phone, their phone line usage could be modeled as a hyper-exponential distribution where there is probability p of them talking on the phone with rate lambda,_1 and probability "q" of them using their internet connection with rate lambda,_2.Properties of the hyper-exponential distribution
Since the expected value of a sum is the sum of the expected values, the expected value of a hyper-exponential random variable can be shown as
:E(X) = int_{-infty}^infty x f(x) dx= p_1int_0^infty xlambda,_1e^{-lambda,_1x} dx+ p_2int_0^infty xlambda,_2e^{-lambda,_2x} dx+ cdots + p_nint_0^infty xlambda,_ne^{-lambda,_nx} dx
::sum_{i=1}^n frac{p_i}{lambda,_i}
and
:E(X^2) = int_{-infty}^infty x^2 f(x) , dx = p_1int_0^infty x^2lambda,_1e^{-lambda,_1x} , dx + p_2int_0^infty x^2lambda,_2e^{-lambda,_2x} , dx+ cdots + p_nint_0^infty x^2lambda,_ne^{-lambda,_nx}, dx,
::sum_{i=1}^n frac{2}{lambda,_i^2}p_i,
from which we can derive the variance.
The
moment-generating function is given by:E(e^{tx}) = int_{-infty}^infty e^{tx} f(x) dx= p_1int_0^infty e^{tx}lambda,_1e^{-lambda,_1x} dx+ p_2int_0^infty e^{tx}lambda,_2e^{-lambda,_2x} dx+ cdots + p_nint_0^infty e^{tx}lambda,_ne^{-lambda,_nx} dx
::sum_{i=1}^n frac{lambda,_i}{lambda_i - t}p_i.
ee also
*
phase-type distribution
Wikimedia Foundation. 2010.