A channel has all input x consisting of the number

A channel has all input x consisting of the numbers +1 and -1 used with the probabilities PX(+1)=PX(-1)=1／2．The output y is the sum of x and an independent noise random variable n with the probability density pn(n)=1／4 for -2≤n≤2 and pn(n)=0 elsewhere.In other words，the conditional probability density of y condition on x is given by pY|X(y|x)=1／4 for -2≤y-x≤2 and pY|X(y|x)=0 elsewhere． (1)Find and sketch the output probability density for the channel． (2)Find I(X；Y)． (3)Suppose that the output is transformed into a new discrete random variable x defined z=1 for y>1；z=0 for -1≤y≤+1；z= -1 for y<-1．Find I(X；Z)and interpret your result． (4)For a discrete X，Y ensemble，let Z be a new ensemble with elements determined by the elements in the Y ensemble，z=z(y)．Show that if P(x|z(y))=P(x|y)，then I(X；Z)=I(X；Y)．
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