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.

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