Mixture Distribution

(c) Tobias Hossfeld (Aug 2021)

This script and the figures are part of the following book. The book is to be cited whenever the script is used (copyright CC BY-SA 4.0):
Tran-Gia, P. & Hossfeld, T. (2021). Performance Modeling and Analysis of Communication Networks - A Lecture Note. Würzburg University Press.

A mixture distribution $A$ is derived from a collection of $k$ other independent random variables $A_1,\dotsc,A_k$. With probability $p_i$, the random variable $A_i$ is chosen, for $i=1,\dotsc,k$. Then, the value of the selected random variable $A_i$ is realized. Thus, the random variable $A$ follows $A_i$ with probability $p_i$. We obtain the following relations.

$ a(t) = \sum_{i=1}^k p_i \cdot a_i(t) \;, \qquad A(t) = \sum_{i=1}^k p_i \cdot A_i(t) \\ E[A] = \sum_{i=1}^k p_i \cdot E[A_i] \;, \quad E[A^n] = \sum_{i=1}^k p_i \cdot E[A_i^n]\\ \Phi_{A}(s) = \int\limits_{0}^\infty e^{-st} a(t) \;dt = \int\limits_{0}^\infty e^{-st} \sum_{i=1}^k p_i \cdot a_i(t) \;dt = \sum_{i=1}^k p_i \cdot \Phi_{A_i}(s) $

Example: Mix of Exponential and Deterministic Distribution

With probability $p=1/2$, the distribution $A$ follows an exponential distribution $X \sim \mathrm{EXP}(\lambda)$ with $\lambda=1$. With probability $p=1/2$, the distribution $A$ follows a deterministic distribution $Y \sim \mathrm{D}(t_0)$ with $t_0=1$. Please note that the sum $Z_0=X+Y$ follows a shifted exponential distribution with PDF $z_0(t) = e^{-(t-0.5)}$ for $t \geq t_0$ and $z_0(t)=0$ for any other $t$. Hence, the weighted sum of distributions ($Z=\frac{1}{2}X+\frac{1}{2}Y$) differs from the mixture distribution $A=\mathrm{MIX}((X,Y),(\frac{1}{2},\frac{1}{2}))$.