hw3 - EE 278 October 7 2009 Statistical Signal Processing...

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Unformatted text preview: EE 278 October 7, 2009 Statistical Signal Processing Handout #4 Homework #3 Due: Wednesday October 14 1. Family planning. Alice and Bob choose a number X at random with equal probability from the set { 2 , 3 , 4 } . If the outcome is X = x , they decide to have children until they have a girl or x children, whichever comes first. Assume that each child is a girl with probability 1 / 2 (independent of the number of children and gender of other children). Let Y be the number of children they will have. a. Find the conditional pmf p Y | X ( y | x ) for all possible values of x and y . b. Find the pmf of Y . 2. First available teller. A bank has two tellers. The service times for tellers 1 and 2 are indepen- dent exponential random variables X 1 ∼ Exp( λ 1 ) and X 2 ∼ Exp( λ 2 ). You arrive at the bank and find that both tellers are busy but nobody else is waiting to be served. You are served by the first available teller who becomes free. What is the probability that you are served by the teller 1? 3. Optical communication channel. The input signal to an optical channel is X = braceleftBigg 1 with probability 1 2 10 with probability 1 2 . The output of the channel is Y , and the conditional pmf of Y given X = a is Poisson with intensity a ; i.e., Y |{ X = 1 } ∼ Poisson(1) and Y |{ X = 10 } ∼ Poisson(10). Show that the MAP rule reduces to D ( y ) = braceleftBigg 1 if y < y * 10 otherwise. Find y * and the corresponding probability of error. 4. Radar signal detection. (Bonus) The received signal S for a radar channel is 0 if there is no target and a random variable X ∼ N (0 ,P ) if there is a target. Both possibilities occur with equal probability. Thusequal probability....
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hw3 - EE 278 October 7 2009 Statistical Signal Processing...

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