Exercise set 1.pdf

# Exercise set 1.pdf - Probability and Stochastic Processes...

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Probability and Stochastic Processes Fall 2017 Exercise set 1 1- The events ࠵? and ࠵? are mutually exclusive and ࠵?(࠵?) ≠ 0 , ࠵?(࠵?) ≠ 0 . Can they be independent? Prove your answer. 2- Let ࠵? ( represent the probability that a randomly selected item from box A is defective; similarly, ࠵? ) represents the probability that a randomly selected item from box B is defective. Boxes ࠵? and ࠵? are identical. Two items are randomly selected from one of the boxes. a) What is the probability that both the items are defective? b) Suppose both the items are found to be defective; what is the probability that they came from box ࠵? ? 3- Consider the following communication channel, also known as the Z-channel: The channel input is either a ′0′ or a ′1′ . A ′0′ is transmitted with probability ࠵? , a ′1′ is transmitted with probability 1 − ࠵? , where 0 ≤ ࠵? ≤ 1 . When a ′0′ is transmitted, with probability 1 it is received as a ′0′ . However, when a ′1′ is

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