ECE PhD QE CNSIP Jan 2002 Problem1 - Rhea

Question

1. (20 pts)

Given two coins; the first coin is fair and the second coin has two heads. One coin is picked at random and tossed two times. It shows heads both times. What is the probability that the coin picked is fair?

Click here to view student answers and discussions

2. (20 pts)

Let $ \mathbf{X}_{t} $ and $ \mathbf{Y}_{t} $ by jointly wide sense stationary continous parameter random processes with $ E\left[\left|\mathbf{X}\left(0\right)-\mathbf{Y}\left(0\right)\right|^{2}\right]=0 $ . Show that $ R_{\mathbf{X}}\left(\tau\right)=R_{\mathbf{Y}}\left(\tau\right)=R_{\mathbf{XY}}\left(\tau\right) $ .

Click here to view student answers and discussions

3. (20 pts)

Let $ \mathbf{X}_{t} $ be a zero mean continuous parameter random process. Let $ g(t) $ and $ w\left(t\right) $ be measurable functions defined on the real numbers. Further, let $ w\left(t\right) $ be even. Let the autocorrelation function of $ \mathbf{X}_{t} $ be $ \frac{g\left(t_{1}\right)g\left(t_{2}\right)}{w\left(t_{1}-t_{2}\right)} $ . From the new random process $ \mathbf{Y}_{i}=\frac{\mathbf{X}\left(t\right)}{g\left(t\right)} $ . Is $ \mathbf{Y}_{t} $ w.s.s. ?

Click here to view student answers and discussions

4. (20 pts)

Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n} $ be i.i.d. random variables with absolutely continuous probability distribution function $ F\left(x\right) $ . Let the random variable $ \mathbf{Y}_{j} $ be the $ j $ -th order statistic of the $ \mathbf{X}_{i} $ 's. that is: $ \mathbf{Y}_{j}=j\text{-th smallest of }\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $ .

(a)

What is another name for the first order statistic?

(b)

What is another name for the n/2 order statistic?

(c)

Find the probability density function of the first order statistic. (You may assume n is odd.)

Click here to view student answers and discussions

5. (20 pts)

Let $ \mathbf{X} $ be a random variable with absolutely continuous probability distribution function. Show that for any $ \alpha>0 $ and any real number $ s $ :$ P\left(e^{s\mathbf{X}}\geq\alpha\right)\leq\frac{\phi\left(s\right)}{\alpha} $ where $ \phi\left(s\right) $ is the moment generating function, $ \phi\left(s\right)=E\left[e^{s\mathbf{X}}\right] $ . Note: $ \phi\left(s\right) $ can be related to the Laplace Transform of $ f_{\mathbf{X}}\left(x\right) $ .

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page