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'''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 [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. <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 [[ECE_PhD_QE_CNSIP_2015_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.2|answers and discussions]]''' | ||
| Line 36: | Line 40: | ||
'''Part 3.''' | '''Part 3.''' | ||
| − | Let <math>X</math> and <math>Y</math> be independent | + | 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 [[ECE_PhD_QE_CNSIP_2015_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.3|answers and discussions]]''' | ||
| Line 42: | Line 50: | ||
'''Part 4.''' | '''Part 4.''' | ||
| − | |||
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.4|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.4|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]] | ||
Revision as of 21:41, 17 February 2019
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.
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Part 2.
A point $ \omega $ is picked at random in the triangle shown here (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.
(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 $.
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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) $).
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Part 4.
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