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.

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