Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2003

Question

Problem 1 (30 points)

i)

Let $\mathbf{X}$ and $\mathbf{Y}$ be jointly Gaussian (normal) distributed random variables with mean $0$ , $E\left[\mathbf{X}^{2}\right]=E\left[\mathbf{Y}^{2}\right]=\sigma^{2}$ and $E\left[\mathbf{XY}\right]=\rho\sigma^{2}$ with $\left|\rho\right|<1$ . Find the joint characteristic function $E\left[e^{i\left(h_{1}\mathbf{X}+h_{2}\mathbf{Y}\right)}\right]$ .

ii)

Let $\mathbf{X}$ and $\mathbf{Y}$ be two jointly Gaussian distributed r.v's with identical means and variances but are not necessarily independent. Show that the r.v. $\mathbf{V}=\mathbf{X}+\mathbf{Y}$ is independeent of the r.v. $\mathbf{W}=\mathbf{X}-\mathbf{Y}$ . Is the same answer true for $\mathbf{A}=f\left(\mathbf{V}\right)$ and $\mathbf{B}=g\left(\mathbf{W}\right)$ where $f\left(\cdot\right)$ and $g\left(\cdot\right)$ are suitable functions such that $E\left[f\left(\mathbf{V}\right)\right]<\infty$ and $E\left[g\left(\mathbf{W}\right)\right]<\infty$ . Given reasons.

iii)

Let $\mathbf{X}$ and $\mathbf{Y}$ be independent $N\left(m,1\right)$ random variables. Show that the sample mean $\mathbf{M}=\frac{\mathbf{X}+\mathbf{Y}}{2}$ is independent of the sample variance $\mathbf{V}=\left(\mathbf{X}-\mathbf{M}\right)^{2}+\left(\mathbf{Y}-\mathbf{M}\right)^{2}$ .

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Problem 2 (35 points)

Consider the stochastic process $\left\{ \mathbf{X}_{n}\right\}$ defined by: $\mathbf{X}_{n+1}=a\mathbf{X}_{n}+b\mathbf{W}_{n} where \mathbf{X}_{0}\sim N\left(0,\sigma^{2}\right)$ and $\left\{ \mathbf{W}_{n}\right\}$ is an i.i.d. $N\left(0,1\right)$ sequence of r.v's independent of $\mathbf{X}_{0}$ .

i)

Show that if $R_{k}=cov\left(\mathbf{X}_{k},\mathbf{X}_{k}\right)$ converges as $k\rightarrow\infty$ , then $\left\{ \mathbf{X}_{k}\right\}$ converges to a w.s.s. process.

ii)

Show that if $\sigma^{2}$ is chosen appropriately and $\left|a\right|<1$ , then $\left\{ \mathbf{X}_{k}\right\}$ will be a stationary process for all $k$ .

iii)

If $\left|a\right|>1$ , show that the variance of the process $\left\{ \mathbf{X}_{k}\right\}$ diverges but $\frac{\mathbf{X}_{k}}{\left|a\right|^{k}}$ converges in the mean square.

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Problem 3 (35 points)

i)

Catastrophes occur at times $\mathbf{T}_{1},\mathbf{T}_{2},\cdots$, where $\mathbf{T}_{i}=\sum_{k=1}^{i}\mathbf{X}_{k}$ where the $\mathbf{X}_{k}$ 's are independent, identically distributed positive random variables. Let $\mathbf{N}_{t}=\max\left\{ n:\mathbf{T}_{n}\leq t\right\}$ be the number of catastrophes which have occurred by time $t$ . Show that if $E\left[\mathbf{X}_{1}\right]<\infty$ then $\mathbf{N}_{t}\rightarrow\infty$ almost surely (a.s.) and $\frac{\mathbf{N}_{t}}{t}\rightarrow\frac{1}{E\left[\mathbf{X}_{1}\right]}$ as $t\rightarrow\infty$ a.s.

ii)

Let $\left\{ \mathbf{X}_{t},t\geq0\right\}$ be a stochastic process defined by: $\mathbf{X}_{t}=\sqrt{2}\cos\left(2\pi\xi t\right)$ where $\xi$ is a $N\left(0,1\right)$ random variable. Show that as $t\rightarrow\infty,\;\left\{ \mathbf{X}_{t}\right\}$ converges to a wide sense stationary process. Find the spectral density of the limit process.

Hint:

Use the fact that the characteristic function of a $N\left(0,1\right)$ r.v. is given by $E\left[e^{ih\mathbf{X}}\right]=e^{-\frac{h^{2}}{2}}$ .

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