Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2006
4
Suppose customer orders arrive according to an i.i.d. Bernoulli random process $ \mathbf{X}_{n} $ with parameter $ p $ . Thus, an order arrives at time index $ n $ (i.e., $ \mathbf{X}_{n}=1 $ ) with probability $ p $ ; if an order does not arrive at time index $ n $ , then $ \mathbf{X}_{n}=0 $ . When an order arrives, its size is an exponential random variable with parameter $ \lambda $ . Let $ \mathbf{S}_{n} $ be the total size of all orders up to time $ n $ .
(a) (20 points)
Find the mean and autocorrelation function of $ \mathbf{S}_{n} $ .
Let $ \mathbf{Y}_{n} $ be the size of an order at time index $ n $ , then $ \mathbf{Y}_{n} $ is a sequence of i.i.d. exponential random variables.
$ \mathbf{S}_{n}=\sum_{k=1}^{n}\mathbf{X}_{n}\mathbf{Y}_{n}. $
$ E\left[\mathbf{S}_{n}\right]=\sum_{k=1}^{n}E\left[\mathbf{X}_{n}\right]E\left[\mathbf{Y}_{n}\right]=\sum_{k=1}^{n}p\cdot\frac{1}{\lambda}=\frac{np}{\lambda}. $
$ R_{\mathbf{S}}\left(n,m\right)=E\left[\mathbf{S}_{n}\mathbf{S}_{m}\right]=\sum_{k=1}^{n}\sum_{l=1}^{m}E\left[\mathbf{X}_{n}\right]E\left[\mathbf{X}_{m}\right]E\left[\mathbf{Y}_{n}\right]E\left[\mathbf{Y}_{m}\right]=\sum_{k=1}^{n}\sum_{l=1}^{m}\frac{p^{2}}{\lambda^{2}}=nm\frac{p^{2}}{\lambda^{2}}. $
(b) (5 points)
Is $ \mathbf{S}_{n} $ a stationary random process? Explain.
• Approach 1: $ \mathbf{S}_{n} $ is not a stationary random process since $ R_{\mathbf{S}}\left(n,m\right) $ does not depend on only $ m-n $ .
• Approach 2: $ \mathbf{S}_{n} $ is not a stationary random process since $ E\left[\mathbf{S}_{n}\right] $ is not constant.
