m
 
(10 intermediate revisions by the same user not shown)
Line 26: Line 26:
 
A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer.
 
A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer.
  
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.1|answers and discussions]]'''
+
:'''Click [[2016CS-1-1|here]] to view student [[2016CS-1-1|answers and discussions]]'''
 
----
 
----
 
'''Part 2.'''
 
'''Part 2.'''
  
A point <math>\omega</math> is picked at random in the triangle shown [https://engineering.purdue.edu/ECE/Academics/Graduates/Archived_QE_August_2016/CS-1?dl=1 here] (all points are equally likely.) let the random variable <math>X(\omega)</math> be the perpendicular distance from <math>\omega</math> to be base as shown in the diagram.  
+
A point <math>\omega</math> is picked at random in the triangle shown below (all points are equally likely.) let the random variable <math>X(\omega)</math> be the perpendicular distance from <math>\omega</math> to be base as shown in the diagram. <br>
 +
https://www.projectrhea.org/rhea/dropbox_/381ea5db244c12bb92e6de3206725a7a/Wan82_CS1_problem.PNG<br>
 +
'''(a)''' Find the cumulative distribution function (cdf) of <math>\mathbf{X}</math>.<br>
 +
'''(b)''' Find the probability distribution function (pdf) of <math>\mathbf{X}</math>.<br>
 +
'''(c)''' Find the mean of <math>\mathbf{X}</math>.<br>
 +
'''(d)''' What is the probability that <math>\mathbf{X}>h/3</math>.<br>
  
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.2|answers and discussions]]'''
+
:'''Click [[2016CS-1-2|here]] to view student [[2016CS-1-2|answers and discussions]]'''
 
----
 
----
 
'''Part 3.'''
 
'''Part 3.'''
  
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>.  
+
Let <math>X</math> and  <math>Y</math> be independent, jointly-distributed Poisson random variables with means with mean <math>\lambda</math> and <math>\mu</math>. Let <math>Z</math> be a new random variable defined as
 +
<br>
 +
<math>Z=X+Y</math> <br>
 +
'''(a)''' Find the probability mass function (pmf) of <math>\mathbf{Z}</math>.<br>
 +
'''(b)''' Show that the conditional probability mass function (pmf) of <math>X</math> conditioned on the event <math>{Z=n}</math> is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters <math>"n"</math> and <math>"p"</math>) required to specify a binomial distribution <math>b(n,p)</math>).<br>
  
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.3|answers and discussions]]'''
+
:'''Click [[2016CS-1-3|here]] to view student [[2016CS-1-3|answers and discussions]]'''
 
----
 
----
 
'''Part 4.'''
 
'''Part 4.'''
  
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.
+
Let <math>X(t)</math> be a wide-sense stationary Gaussian random process with mean <math>\mu_x</math> and autocorrelation function <math>R_xx(\tau)</math>. Let <br>
 
+
<math>Y(t)=c_1X(t)-c_2X(t-T)</math>,<br>
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.4|answers and discussions]]'''
+
where <math>c_1,c_2</math> and <math>T</math> are real numbers. What is the probability that <math>Y(t)</math> is less than or equal to a real number <math>/\gamma?</math> Express your answer in terms of <math>c_1,c_2,\mu_x,\sigma_x^2</math>, and <math>R_xx(\tau), \gamma</math> and the "phi function"<br>
 +
<math>\Phi(x)=\int_{-\infty}^{x} \dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2} dz</math><br>
 +
:'''Click [[2016CS-1-4|here]] to view student [[2016CS-1-4|answers and discussions]]'''
 
----
 
----
 
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]
 
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]

Latest revision as of 15:33, 19 February 2019


ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2016



Question

Part 1.

A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer.

Click here to view student answers and discussions

Part 2.

A point $ \omega $ is picked at random in the triangle shown below (all points are equally likely.) let the random variable $ X(\omega) $ be the perpendicular distance from $ \omega $ to be base as shown in the diagram.
Wan82_CS1_problem.PNG
(a) Find the cumulative distribution function (cdf) of $ \mathbf{X} $.
(b) Find the probability distribution function (pdf) of $ \mathbf{X} $.
(c) Find the mean of $ \mathbf{X} $.
(d) What is the probability that $ \mathbf{X}>h/3 $.

Click here to view student answers and discussions

Part 3.

Let $ X $ and $ Y $ be independent, jointly-distributed Poisson random variables with means with mean $ \lambda $ and $ \mu $. Let $ Z $ be a new random variable defined as
$ Z=X+Y $
(a) Find the probability mass function (pmf) of $ \mathbf{Z} $.
(b) Show that the conditional probability mass function (pmf) of $ X $ conditioned on the event $ {Z=n} $ is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters $ "n" $ and $ "p" $) required to specify a binomial distribution $ b(n,p) $).

Click here to view student answers and discussions

Part 4.

Let $ X(t) $ be a wide-sense stationary Gaussian random process with mean $ \mu_x $ and autocorrelation function $ R_xx(\tau) $. Let
$ Y(t)=c_1X(t)-c_2X(t-T) $,
where $ c_1,c_2 $ and $ T $ are real numbers. What is the probability that $ Y(t) $ is less than or equal to a real number $ /\gamma? $ Express your answer in terms of $ c_1,c_2,\mu_x,\sigma_x^2 $, and $ R_xx(\tau), \gamma $ and the "phi function"
$ \Phi(x)=\int_{-\infty}^{x} \dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2} dz $

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page

Alumni Liaison

Questions/answers with a recent ECE grad

Ryne Rayburn