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.

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**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 $ .

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**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.

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