Appendix D

Spectral Appendix D PhL 5.12.13 Appendix D: Probability Theory: how α and β are related to μ and σ 1 (a) Random Variables X and Y 1 (b) Application to the Pulse Train Ensemble 3 (c) Main conclusions for the pulse train ensemble 4 Appendix D: Probability Theory: how α and β are related to μ and σ We first review some of the basics of random variables X and Y: expectation values, means, covariance, variance and standard deviation. We do this in a context in which each variable takes on a set of discrete values rather than continuous values. We then translate our various facts into the language of X,Y → Ym Yn where the latter are random variables associated with the amplitudes at positions m and n of a pulse train. The labels m and n add a notational complexity that is a little difficult to comprehend unless things are written out explicitly, which is done below. (a) Random Variables X and Y If X and Y are random variables which can take discrete values xk and yk, one can express the mean value of each variable as μx = E(X) = Σk xk px(xk) μy = E(Y) = Σk' yk' py(yk') (D.1) where E(X) means the expectation value of X. The sum Σk is over all values in the set of values that X can take, and Σk' is over all values that Y can take. Probability px(xk) is the probability that X takes the value xk and similarly for py(yk'). If we have a very large ensemble of I systems we can measure the above quantities in this manner, μx = E(X) = (1/I) Σi=1I x(i) μy = E(Y) = (1/I) Σi=1I y(i) (D.2) where i labels a particular system in the ensemble, and x(i) and y(i) are the values X and Y take in ensemble system i. The probability functions like px(xk) do not appear in (D.2), but they are present in spirit, being embedded in the "experimental data" like x(i). The expectation value E(XY) is defined by E(XY) = ΣkΣk' xk yk' pxy(xk,yk') (D.3) where now pxy(xk,yk') is the joint probability that X has value xk when Y has the value yk'. One would measure E(XY) in this manner, E(XY) = (1/I) Σi=1I x(i) y(i) . (D.4) If the random variables X and Y are independent (no correlation), then pxy(xk,yk') = px(xk) py(yk') (D.5) and then it is obvious from the above definitions of E(XY), E(X) and E(Y) that E(XY) = E(X)E(Y) . (D.6) The covariance of X and Y is defined as, cov(X,Y) = E( [X - μx] [Y - μy] ) = ΣkΣk' (xk - μx) (yk'- μy) pxy(xk,yk') (D.7) and it could be measured for an ensemble in this way cov(X,Y) = (1/I) Σi=1I (x(i) - μx) (y(i)- μy) (D.8) where μx and μy would be the measured values of E(X) and E(Y) as shown above. If X and Y are independent variables then we use (D.5) in (D.7) to find that cov(X,Y) = E( [X - μx] [Y - μy] ) = E(X - μx) E(Y - μy) . (D.9) In the special case that X and Y are the same variable, we write cov(X,Y) = var(X), the variance of X, which is also defined as the square of the standard deviation σx, var(X) ≡ σx2 ≡ cov(X,X) = ΣkΣk' (xk - μx) (xk'- μx) pxx(xk,xk') . (D.10) But pxx(xk,xk') = δk.k'px(xk) because if X has value xk, it cannot also have a different value xk', so pxx(xk,xk') = δk,k' px(xk) (D.11) and then var(X) ≡ σx2 ≡ cov(X,X) = Σk (xk - μx)2 px(xk) (D.12) with experimental measurement var(X) ≡ σx2 ≡ cov(X,X) = (1/I) Σi=1I (x(i) - μx)2 . (D.13) Finally, we note that cov(X,Y) = E( [X - μx] [Y - μy]) = E(XY) - μxE(Y) - μyE(X) + μxμyE(1) = E(XY) - μxμy - μyμx + μxμy = E(XY) - μxμy . (D.14) When X and Y are the same random variable this says var(X) ≡ σx2 ≡ cov(X,X) = E(X2) - μx2 . (D.15) (b) Application to the Pulse Train Ensemble In the above we set X = Ym = random variable associated with the amplitudes ym(i) at pulse location m in pulse train Y = Yn = random variable associated with the amplitudes yn(i) at pulse location n in pulse train Things are a little complicated due to the extra labels m and n, but we can nevertheless carefully translate all the X and Y equations above. Also, we shall add the following extra notation for an expectation value to be consistent with our earlier work, <x> ≡ E(X) <xy> ≡ E(XY) <x2> ≡ E(X2) . (D.16) The first pair of equations above translate into the following: μym = E(Ym) = <ym> = Σk (ym)k pym[(ym)k] = Σk (ym)k p[(ym)k] = μ μyn = E(Yn) = <yn> = Σk (yn)k pyn[(yn)k] = Σk (ym)k p[(ym)k] = μ . (D.17) Here (ym)k is the kth value in the set of all possible amplitudes that position m in the pulse train can have. For example, we might have (ym)1 = A and (ym)2 = B. For the pulse train, the set of allowed amplitude values is the same for any pulse in the train, and the probability distribution is also the same, yielding the simplifications shown above. In particular, the means μym and μyn don't depend on position m and n, so we can just call both these means μ. Here is how the above means would be measured for an ensemble, μ = μym = E(Ym) = <ym>= (1/I) Σi=1I ym(i) μ = μyn = E(Yn) = <yn>= (1/I) Σi=1I yn(i) . (D.18) Next, for E(XY) we translate (D.3) to get , E(YmYn) = <ymyn> = ΣkΣk' (ym)k (yn)k' pym,yn[(ym)k, (yn)k'] E(YmYn) = <ymyn> = (1/I) Σi=1I ym(i) yn(i) . (D.19) If Ym and Yn are uncorrelated (which requires that m ≠n ) then in analogy with (D.5), pym,yn[(ym)k, (yn)k'] = pym[(ym)k] pyn[(yn)k] = p[(ym)k] p[(yn)k] (D.20) and as in (D.6) we find that, in the case of no correlation, E(YmYn) = <ymyn> = E(Ym) E(Yn) = <ym><yn> . m ≠ n (D.21) Obviously if m = n, then Ym and Yn cannot be "uncorrelated" since they are the same random variable. If m = n, then we have instead this version of (D.11), pym,ym[(ym)k, (ym)k'] = δk,k' pym[(ym)k] = δk,k' p[(ym)k] (D.22) and then E(Ym2) = <ym2> = Σk [(ym)k]2 p[(ym)k] (D.23) which in experiment (over the ensemble) would be measured as E(Ym2) = <ym2> = (1/I) Σi=1I (ym(i))2 . (D.24) The covariance equations translated from (D.7) and (D.8) become cov(Ym, Yn) = E( [Ym - μ] [Yn - μ] ) = ΣkΣk' ((ym)k - μ) ((ym)k'- μ) pym,yn[(ym)k, (yn)k'] cov(Ym, Yn) = (1/I) Σi=1I (ym(i) - μ) (yn(i)- μ) . (D.25) The variance (and standard deviation) translations from (D.12) and (D.13) are. var(Ym) ≡ σ2 ≡ cov(Ym, Ym) = Σk [(ym)k - μ]2 p[(ym)k] var(Ym) ≡ σ2 ≡ cov(Ym, Ym) = (1/I) Σi=1I (ym(i) - μ)2 . (D.26) Finally, (D.14) and (D.15) become, cov(Ym, Yn) = E(Ym Yn) - μyxμyn = E(Ym Yn) - μ2 var(Ym) ≡ σ2 ≡ cov(Ym, Ym) = E(Ym2) - μ2 = <ym2> - μ2 => <ym2> = σ2+μ2 . (D.27) (c) Main conclusions for the pulse train ensemble The following quantities are measured from the pulse train ensemble as follows, <ym> = (1/I) Σi=1I ym(i) = μ (D.18) <ymyn> = (1/I) Σi=1I ym(i) yn(i) any m and n (D.19) <ym2> = (1/I) Σi=1I [ym(i]]2 = σ2 + μ2 (D.27) (D.28) where we show how each quantity is related to symbols μ and σ. When the amplitudes at different locations in the pulse train are uncorrelated, we can further write <ymyn> = <ym><yn> = μ2 m ≠n . (D.21) (D.29) We can now restate equation (35.6) in this more statistically-oriented manner, <ym> = μ = [p]A + [1-p]B n ≠ m α ≡ <ymyn> = μ2 = { [p]A + [1-p]B }2 = [pp] AA + [p(1-p)] 2AB + [(1-p)(1-p)]BB n = m β ≡ <ym2> = [p]AA + [(1-p)] BB = σ2 + μ2 (D.30) which is to say α = μ2 β = μ2 + σ2 = α + σ2 σ2 = (β-α) . (D.31) Using our expressions for α and β we can compute σ2: which says σ2 = variance = [ p(1-p)] (A-B)2 . (D.32) If p = 0 or p = 1, this variance vanishes as one would expect.