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==Question== | ==Question== | ||
| − | ''' | + | '''1. (30 Points)''' |
| − | + | Assume that <math class="inline">\mathbf{X}</math> is a binomial distributed random variable with probability mass function (pmf) given by <math class="inline">p_{n}\left(k\right)=\left(\begin{array}{c} | |
| + | n\\ | ||
| + | k | ||
| + | \end{array}\right)p^{k}\left(1-p\right)^{n-k}\;,\qquad k=0,1,2,\cdots,n</math> where <math class="inline">0<p<1</math> . | ||
| + | |||
| + | '''(a)''' | ||
| + | |||
| + | Find the characteristic function of <math class="inline">\mathbf{X}</math> . (You must show how you derive the characteristic function.) | ||
| + | |||
| + | '''(b)''' | ||
| + | |||
| + | Compute the standard deviation of <math class="inline">\mathbf{X}</math> . | ||
| + | |||
| + | '''(c)''' | ||
| + | |||
| + | Find the value or values of <math class="inline">k</math> for which <math class="inline">p_{n}\left(k\right)</math> is maximum, and express the answer in terms of <math class="inline">p</math> and <math class="inline">n</math> . Give the most complete answer to this question that you can. | ||
:'''Click [[ECE_PhD_QE_CNSIP_2005_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2005_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2005_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2005_Problem1.1|answers and discussions]]''' | ||
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| − | ''' | + | '''2. (30 Points)''' |
| + | |||
| + | Let <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots</math> be a sequence of binomially distributed random variables, with <math class="inline">\mathbf{X}_{n}</math> having probability mass function <math class="inline">p_{n}\left(k\right)=\left(\begin{array}{c} | ||
| + | n\\ | ||
| + | k | ||
| + | \end{array}\right)p_{n}^{k}\left(1-p_{n}\right)^{n-k}\;,\qquad k=0,1,2,\cdots,n,</math> where <math class="inline">0<p_{n}<1</math> for all <math class="inline">n=1,2,3,\cdots</math> . Show that if <math class="inline">np_{n}\rightarrow\lambda\text{ as }n\rightarrow\infty,</math> then the random sequence <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots</math> converges in distribution to a Poisson random variable having mean <math class="inline">\lambda</math> . | ||
| − | |||
:'''Click [[ECE_PhD_QE_CNSIP_2005_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2005_Problem1.2|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2005_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2005_Problem1.2|answers and discussions]]''' | ||
---- | ---- | ||
| − | ''' | + | '''3. (40 Points)''' |
| − | + | Consider a homogeneous Poisson point process with rate <math class="inline">\lambda</math> and points (event occurrence times) <math class="inline">\mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{n},\cdots</math> . | |
| − | + | '''(a)''' | |
| − | + | ||
| − | + | ||
| − | + | Derive the pdf <math class="inline">f_{k}\left(t\right)</math> of the <math class="inline">k</math> -th point <math class="inline">\mathbf{T}_{k}</math> for arbitrary <math class="inline">k</math> . | |
| − | :'''Click [[ECE_PhD_QE_CNSIP_2005_Problem1. | + | '''(b)''' |
| + | |||
| + | What kind of distribution does <math class="inline">\mathbf{T}_{1}</math> have? | ||
| + | |||
| + | '''(c)''' | ||
| + | |||
| + | What is the conditional pdf of <math class="inline">\mathbf{T}_{k}</math> given <math class="inline">\mathbf{T}_{k-1}=t_{0}</math> , where <math class="inline">t_{0}>0</math> ? (You can give the answer without derivation if you know it.) | ||
| + | |||
| + | '''(d)''' | ||
| + | |||
| + | Suppose you have a random number generator that produces independent, identically distributed (i.i.d. ) random variables <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots</math> that are uniformaly distributed on the interval <math class="inline">\left(0,1\right)</math> . Explain how you could use these to simulate the Poisson points <math class="inline">\mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{n},\cdots</math> describe above. Provide as complete an explanation as possible. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2005_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2005_Problem1.3|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 00:54, 10 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2005
Question
1. (30 Points)
Assume that $ \mathbf{X} $ is a binomial distributed random variable with probability mass function (pmf) given by $ p_{n}\left(k\right)=\left(\begin{array}{c} n\\ k \end{array}\right)p^{k}\left(1-p\right)^{n-k}\;,\qquad k=0,1,2,\cdots,n $ where $ 0<p<1 $ .
(a)
Find the characteristic function of $ \mathbf{X} $ . (You must show how you derive the characteristic function.)
(b)
Compute the standard deviation of $ \mathbf{X} $ .
(c)
Find the value or values of $ k $ for which $ p_{n}\left(k\right) $ is maximum, and express the answer in terms of $ p $ and $ n $ . Give the most complete answer to this question that you can.
- Click here to view student answers and discussions
2. (30 Points)
Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots $ be a sequence of binomially distributed random variables, with $ \mathbf{X}_{n} $ having probability mass function $ p_{n}\left(k\right)=\left(\begin{array}{c} n\\ k \end{array}\right)p_{n}^{k}\left(1-p_{n}\right)^{n-k}\;,\qquad k=0,1,2,\cdots,n, $ where $ 0<p_{n}<1 $ for all $ n=1,2,3,\cdots $ . Show that if $ np_{n}\rightarrow\lambda\text{ as }n\rightarrow\infty, $ then the random sequence $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots $ converges in distribution to a Poisson random variable having mean $ \lambda $ .
- Click here to view student answers and discussions
3. (40 Points)
Consider a homogeneous Poisson point process with rate $ \lambda $ and points (event occurrence times) $ \mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{n},\cdots $ .
(a)
Derive the pdf $ f_{k}\left(t\right) $ of the $ k $ -th point $ \mathbf{T}_{k} $ for arbitrary $ k $ .
(b)
What kind of distribution does $ \mathbf{T}_{1} $ have?
(c)
What is the conditional pdf of $ \mathbf{T}_{k} $ given $ \mathbf{T}_{k-1}=t_{0} $ , where $ t_{0}>0 $ ? (You can give the answer without derivation if you know it.)
(d)
Suppose you have a random number generator that produces independent, identically distributed (i.i.d. ) random variables $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots $ that are uniformaly distributed on the interval $ \left(0,1\right) $ . Explain how you could use these to simulate the Poisson points $ \mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{n},\cdots $ describe above. Provide as complete an explanation as possible.
- Click here to view student answers and discussions
