By E. Poisson

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**Example text**

Into the geometric picture. J , not over [a, b 1 . Q is fmite-dimensional. Thus has six points, = {1,2,3,4,5,6 }, corresponding to a die-tossing suppose experiment. , X( w) :;::: 1 when w is even, and X( w) = 0 when w is odd. , P( w) = 1/6 for all w . ) can be defmed as J n n e (X, Y}=l: X(w)Y(w)P(w) Wo1 which is the exact analog of the inner-product ( • }2 we defmed in n-space. I X(w)Y(w)P(w) = EXY 28 which is of course consistent with the defmition we used so far. Moreover, since EXY is usually calculated via the joint density function of X and Y rather than via the usually unknown probability measure P(w) on the usually unknown space Q, we note that in many probabilistic calculations we rarely need to explicitly bring in a probability space.

Is desired. However, in certain cases, h(t 0 ,'t") will turn out to depend only on the difference t 0 - t. , with an abuse of notation Ryy(T, t) = Ryy(r- t) Rzy(r, t) = Rzy (r- t). The physical meaning of this assumption need not be explored here. The mathematical consequence will be that the integral equation will reduce to a form that can be easily solved by Fourier transformation. The equation becomes Rzy(t 0 -t)=_[RYY(r-t)h(t 0 ,r) dr, -oo~t~oo We can simplify this by certain naturally suggested changes of variable.

We can readily calculate E e2 • mm = N0 f c ~ 2A~ No - 1n ( 1 + [ 1 - 2A N 0 Now, for low SNR (A ftxed, N0 --+m) we see that (some care is needed- we must keep at least two terms in the expansion of ln(l + 2A/N 0 )) 2 • -+ Af , E e mm c while for high SNR (A--+m, No fixed), 2 • -+ N f . E e mm 0 c These formulas have simple interpretations. Afc turns out to be the variance of the signal process z (Rz(O) = Sz(f) df), which is what we would expect to get since Hopt(f)--+0 as N0 --+m and therefore Z-+0.

### A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson

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