Difference between revisions of "ECE-QE CS1-2015" - Rhea

Latest revision as of 23:17, 3 December 2015

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2015

Question

Part 1.

If $$ X $$ and $$ Y $$ are independent Poisson random variables with respective parameters $\lambda_1$ and $\lambda_2$ , calculate the conditional probability mass function of $$ X $$ given that $$ X+Y=n $$ .

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Part 2.

Let $Z(t), t\ge 0$ , be a random process obtained by switching between the values 0 and 1 according to the event times in a counting process $$ N(t) $$ . Let $$ P(Z(0)=0)=p $$ and

$P(N(t)=k) = \frac{1}{1+\lambda t}(\frac{\lambda t}{1+\lambda t})^k$

for $$ k = 0, 1, ... $$ . Find the pmf of $$ Z(t) $$ .

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Part 3.

Let $$ X $$ and $$ Y $$ be independent identically distributed exponential random variables with mean $\mu$ . Find the characteristic function of $$ X+Y $$ .

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Part 4.

Consider a sequence of independent and identically distributed random variables $$ X_1,X_2,... X_n $$ , where each $$ X_i $$ has mean $\mu = 0$ and variance $\sigma^2$ . Show that for every $$ i=1,...,n $$ the random variables $$ S_n $$ and $$ X_i-S_n $$ , where $S_n=\sum_{j=1}^{n}X_j$ is the sample mean, are uncorrelated.

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@@ Line 32: / Line 32: @@
 Let <math>Z(t), t\ge 0</math>, be a random process obtained by switching between the values 0 and 1 according to the event times in a counting process <math>N(t)</math>. Let <math>P(Z(0)=0)=p</math> and
-<math>Cov(X_i,X_j) = \left\{ \begin{array}{ll}
+<math>P(N(t)=k) =
- {\sigma}^2, & i=j\\
+\frac{1}{1+\lambda t}(\frac{\lambda t}{1+\lambda t})^k</math>
- \rho{\sigma}^2, & |i-j|=1\\
-, & otherwise
-  \end{array} \right.</math>
-Suppose we take 2 consecutive samples from this sequence to form a vector <math>X</math>, which is then linearly transformed to form a 2-dimensional random vector <math>Y=AX</math>. Find a matrix <math>A</math> so that the components of <math>Y</math> are independent random variables You must justify your answer.
+for <math> k = 0, 1, ...</math>. Find the pmf of <math> Z(t)</math>.
-:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.2|answers and discussions]]'''
+:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.2|answers and discussions]]'''
 ----
 '''Part 3.'''
-Let <math>X</math> be an exponential random variable with parameter <math>\lambda</math>, so that <math>f_X(x)=\lambda{exp}(-\lambda{x})u(x)</math>. Find the variance of <math>X</math>. You must show all of your work.
+Let <math>X</math> and  <math>Y</math> be independent identically distributed exponential random variables with mean <math>\mu</math>. Find the characteristic function of <math>X+Y</math>.
-:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.3|answers and discussions]]'''
+:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.3|answers and discussions]]'''
 ----
 '''Part 4.'''
-Consider a sequence of independent random variables <math>X_1,X_2,...</math>, where <math>X_n</math> has pdf
+Consider a sequence of independent and identically distributed random variables <math>X_1,X_2,... X_n</math>, where each <math>X_i</math> has mean <math>\mu = 0</math> and variance <math>	\sigma^2</math>. Show that for every <math>i=1,...,n</math> the random variables <math>S_n</math> and <math>X_i-S_n</math>, where <math>S_n=\sum_{j=1}^{n}X_j</math> is the sample mean, are uncorrelated.
-<math>\begin{align}f_n(x)=&(1-\frac{1}{n})\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{1}{2\sigma^2}(x-\frac{n-1}{n}\sigma)^2]\\
+:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.4|answers and discussions]]'''
-&+\frac{1}{n}\sigma exp(-\sigma x)u(x)\end{align}</math>.
-Does this sequence converge in the mean-square sense? ''Hint:'' Use the Cauchy criterion for mean-square convergence, which states that a sequence of random variables <math>X_1,X_2,...</math> converges in mean-square if and only if <math>E[|X_n-X_{n+m}|] \to 0</math> as <math>n \to \infty</math>, for every <math>m>0</math>.
-:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.4|answers and discussions]]'''
 ----
 [[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]

Difference between revisions of "ECE-QE CS1-2015" - Rhea

Latest revision as of 23:17, 3 December 2015

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