h11 - Detection& Estimation Theory Course No 16:332:549...

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Unformatted text preview: Detection & Estimation Theory Course No: 16:332:549 Solutions to Homework 1 1. Let the event A = { k heads in a specific order } Let the event C i = { The event of selecting the i th coin } Then P ( C i ) = 1 /m . Further, P ( A | C i ) = p k i (1 − p i ) n − k . Then q r = P ( C r | A ) follows by Bayes rule and the total probability theorem. 2. (a) The characteristic function of X i,n is M X i,n ( u ) = E [exp( juX i,n )] = (1 − λ/n ) + exp( ju ) λ/n = 1 + λ n [exp( ju ) − 1] Therefore, M Y n ( u ) = { 1 + λ n [exp( ju ) − 1] } n (b) lim n →∞ M Y n ( u ) = exp( λ (exp( ju ) − 1)), which implies lim n →∞ Y n is a Poisson random variable with mean and variance λ . (This follows from the fact that the mapping from a characteristic function to a distribution is 1:1) 3. We first observe that the signals { s i ( t ) } i = 1 , 2 , 3 are linearly independent. The energy of signal s 1 ( t ) is given as E 1 = Z T s 2 1 ( t ) dt = 4 , where T = 3. Therefore, the first basis function is φ 1 ( t ) = s 1 ( t ) √ E 1 = ( 1 , ≤ t ≤ 1 , otherwise Based on the definition of the coefficients as s ij = Z T s i ( t ) φ j ( t ) dt, (1) we can find that s 21 = − 4....
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h11 - Detection& Estimation Theory Course No 16:332:549...

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