Hyper-exponential distribution

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In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by:

f_X(x) = \sum_{i=1}^n f_{Y_i}(y) p_i

Where Yi is an exponentially distributed random variable with rate parameter \lambda\,_i, and pi 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 the Hypoexponential distribution, which has a coefficient of variation less than one. While the exponential distribution is the continuous analogue of the geometric distribution, the hyper-exponential distribution is not analogous to the hypergeometric distribution.

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

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_1\int_0^\infty x\lambda\,_1e^{-\lambda\,_1x} + p_2\int_0^\infty x\lambda\,_2e^{-\lambda\,_2x} + \cdots + p_n\int_0^\infty x\lambda\,_ne^{-\lambda\,_nx}
= \sum_{i=1}^n \frac{p_i}{\lambda\,_i}

and

E(X^2) = \int_{-\infty}^\infty x^2 f(x) dx= p_1\int_0^\infty x^2\lambda\,_1e^{-\lambda\,_1x} + p_2\int_0^\infty x^2\lambda\,_2e^{-\lambda\,_2x} + \cdots + p_n\int_0^\infty x^2\lambda\,_ne^{-\lambda\,_nx}
= \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_1\int_0^\infty e^{tx}\lambda\,_1e^{-\lambda\,_1x} + p_2\int_0^\infty e^{tx}\lambda\,_2e^{-\lambda\,_2x} + \cdots + p_n\int_0^\infty e^{tx}\lambda\,_ne^{-\lambda\,_nx}
= \sum_{i=1}^n \frac{\lambda\,_i}{\lambda_i - t}p_i

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