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| − | Let <math>X</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 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]]''' | :'''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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'''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> | + | 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]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.4|answers and discussions]]''' | ||
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[[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 00:17, 4 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.
- Click here to view student answers and discussions
