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1assignment

# 1assignment - Assignment 1 Problems 1 A defective coin...

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Assignment 1 Problems: 1. A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF f P ( p ) = braceleftBigg pe p , p [0 , 1] , 0 , otherwise . A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent. (a) Find the probability that a coin toss results in heads. (b) Given that a coin toss resulted in heads, find the conditional PDF of P . (c) Given that a first coin toss resulted in heads, find the conditional probability of heads on the next toss. 2. Show that for real x ( t ), we have x e ( t ) = a 0 2 + summationdisplay n =1 a n cos parenleftbigg 2 π n T 0 t parenrightbigg x o ( t ) = summationdisplay n =1 b n sin parenleftbigg 2 π n T 0 t parenrightbigg where x e ( t ) and x o ( t ) denote the even and odd parts of x ( t ), defined as x e ( t ) = x ( t ) + x ( - t ) 2 x o ( t ) = x ( t ) - x ( - t ) 2 . 3. Prove the scaling property of the Fourier transform. 4. Let x ( t ) and y ( t ) be two periodic signals with period T 0 , and let x n and y n denote the Fourier series coefficients of these two signals. Show that

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