old App G seq section REVD

temp storage PhL 7.29.13 Here I am just storing an old and ill-formed App G section on sequences. (g) Experiments with Sequences Our pulse trains have a set of amplitudes {yi} which form a sequence. If the sequence has N elements, then a sequence can be written {y1, y2.....yN}. When many pulse trains are generated in some line code (perhaps AMI), one finds that the values of yn in position n of the sequence can vary, and that there is some probability distribution associated with the parameter yn. An experiment could be the generation of such a sequence by some apparatus according to some set of rules (those for an AMI line code, say), and after many experiments one could find a probability distribution for parameter yn. Thus, the position n in the sequence is associated with a random variable we must call Yn which takes values yn. This notation leads to some confusion. When we had a random variable H which took values h, we might have enumerated the set of h values of the sample space as {hk} ! To maintain clarity, if we have random variable Yn which takes values yn, we should enumerate those values as (yn)k. Here, yn is the name of a parameter, just as h was the name of a parameter. Thus, to write down a specific sequence "i" we really should say {(y1)i, (y2)i.....(yN)i} = sequence "i" or {y1(i), y2(i)..... y2(i)} = sequence "i" where (y2)i = y2(i) is some specific symbol value, perhaps (y2)i = y2(i) = 5. We used the latter notation when talking about an ensemble of pulse trains in Section ***. On the other hand if we want to represent an arbitrary sequence whose elements are variables, we would write the sequence as {y1, y2.....yN} . In our first experiment, our apparatus makes a single sequence we will call sequence 1 " : {(y1)1, (y2)1}. The elements of this sequence are the amplitudes of a very short pulse train. The symbols are taken from the finite set {1,2,3,4,5,6}. The experiment creates values for variables y1 and y2 according to some probabilistic algorithm so there is some pmfY1(y1) and some pmfY2(y2) . Then Y1 and Y2 are random variables, and so is Z = Y1Y2. There will be some pmfZ(z) = Σy1,y2 pmfY1Y2(y1,y2) . We now interrupt this experiment to get another fact on the table: Fact: The expected value of y1y2 resulting from running this experiment many times will be <y1y2> = Σi=16 Σj=16 yiyj pmfY1Y2(yi,yj) . <y1y2> = Σi=16 Σj=16 yiyj pmf(y1 = yi,y2 = yj) . In the case that Y1 and Y2 are uncorrelated and pmfY1(w) = pmfY2(w) = 1/6, a plot of pmfZ(z) for z = y1y2 will look exactly like the last plot shown above. The expected value of y1y2 will be <y1y2> = Σi=16 Σj=16 yiyj pmfY1(yi) pmfY2(yj) = (1/36) Σi=16 Σj=16 yiyj = (1/36)( Σi=16yi)2 = (1/36) (1 + 2 + 3.. + 6)2 = (1/36) (21)2 = 49/4 = 12.25 Alternatively, using the p(z) values which make up pmfZ(z) shown above, In a different experiment with the same pulse train we can study the random variable Y12 which relates to just the first position in the pulse train. The sample space of possible outcomes is {1,4,9,16,25,36} . We find in general that, for the simple case, <y12> = Σi=16 yi2 pmfY1(yi) = (1/6) Σi=16 yi2 = (1/6)( 1+4+9+16+25+36 ) = 91/6 = 15.17 and the same for <y22> . In our second experiment, we consider an arbitrary pulse train with amplitudes {y1,y2.....yN} that is created by our apparatus with some pdfY1Y2Y3...YN(y1,y2.....yN). For some particular pulse train positions i and j we have the following expected value, <yiyj> = Σa=16 Σb=16 ...... yiyj pdfY1Y2Y3...YN(a,b...i...j...) where the indices a,b,c.... sum over all the yk positions in pdfY1Y2Y3...YN(y1,y2.....yN) other than yi and yj. TBC. Notice that when you run this experiment a million times, you create an ensemble of pulse trains, so I have finally arrived at a place I have been looking for. Also <yi2> = Σa=16 Σb=16 ...... yiyj pdfY1Y2Y3...YN(a,b...i...i...) If everything is independent, then <yiyj> = Σa=16 Σb=16 ...... yiyj pdf(a)pdf(b)....pdf(yi)...pdf(yj),,, = yiyj.pdf(yi) .pdf(yj) <yi2> = yi2 pdf(yi) .pdf(yi) // by same argument This is for a particular position, seems strange there is no sum!