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Midterm Solutions Spring 2011

# Midterm Solutions Spring 2011 - EL6303 Midterm Exam...

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EL6303 Midterm Exam Solutions Spring 2011 1. Three types of messages arrive at a message center: “high priority,” denoted by the letter H, “normal priority,” denoted by the letter N, and “low priority,” denoted by the letter L. Assume that 1 . 0 ) ( = Η Π , 4 . 0 ) ( = Ν Π , 5 . 0 ) ( = Λ Π a) Find the probability that out of 20 messages, 5 are high priority, 10 are normal priority, and 5 are low priority. This is generalized bernouli trials and the answer is 5 10 5 ) 5 . 0 ( ) 4 . 0 ( ) 1 . 0 ( ! 5 ! 10 ! 5 ! 20 b) Suppose that messages arrive one at a time. Find the probability that the first high priority message arrives at the th k message. This event looks like { } H H H H H H H H , , , . . . , , , , , which is 1 - κ s H ' followed by one H . Since they are independent we get { } { } { } ) 1 . 0 ( ) 1 . 0 1 ( , , , . . . , , , , , 1 1 - - - = = k k H P H P H H H H H H H H P 2. X is a uniform random variable on the interval (0,2). Let ) ( X g Y = where < = 1 , 1 0 , ) ( 2 x x x x x g (a) Find and sketch ) ( y F Y . For 1 0 < < ψ { } { } { } ) ( ) ( 2 y F y X P y X P y Y P y F X Y = £ = £ = £ = But it is easy to find that

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Midterm Solutions Spring 2011 - EL6303 Midterm Exam...

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