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| + | [[Category:random variables]] | ||
| + | [[Category:probability]] | ||
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[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
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| − | CS | + | Communication, Networking, Signal and Image Processing (CS) |
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| + | Question 1: Probability and Random Processes | ||
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| + | August 2015 | ||
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| − | '''1.''' | + | ---- |
| + | ==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 [[ECE_PhD_QE_CNSIP_2015_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.1|answers and discussions]]''' | ||
---- | ---- | ||
| − | '''2.''' | + | '''Part 2.''' |
| + | 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. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.2|answers and discussions]]''' | ||
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| − | '''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>. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.3|answers and discussions]]''' | ||
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| − | '''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. | ||
| + | :'''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 22:22, 17 February 2019
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2015
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.
- Click here to view student answers and discussions
Part 3.
Let $ X $ and $ Y $ be independent identically distributed exponential random variables with mean $ \mu $. Find the characteristic function of $ X+Y $.
- Click here to view student answers and discussions
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.
- Click here to view student answers and discussions
