Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2016
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.
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Part 2.
A point $ \omega $ is picked at random in the triangle shown here (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.
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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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