# An introduction to bundles, connections, metrics and by Taubes C.H. PDF By Taubes C.H.

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7. 1-2. Ee means, of course, ixt E (cos xt+i sin xt) = E (cos xt)+iE (sin xt). 7. In the proof outlined it has tacitly been assumed that ii(|jt|) exists: actually this hypothesis is superfluous—the existence of Ex suffices. See Cramer . 3 2 12. F U R T H E R PROBABILITY DISTRIBUTIONS 1 2 . 1 . Three i m p o r t a n t variates derived from the n o r m a l variate are denoted by Xl t a n d F . n mn T h e variate % is defined in the following fashion. Let x independent standardized n o r m a l variates.

Let k a n d k be two statistically independent discrete r a n d o m variables defined o n t h e result space { 0 , 1 , 2, . . } . T h e n (k -\-k ) is a discrete r a n d o m variable defined o n t h e same result space, a n d its generating function is t h e product of t h e generating functions of k a n d k . 1 2 x x Proof. 3, since with k a n d k also t a n d t *, 0 < 111 < 1, are statistically independent. 1) is most conveniently obtained by means of this proposition. kl x k 2 9 . 5 . Proposition.

Is the product of the densities of x y, z . . 5). 1. The symbol "Prob (. . )" is used to denote the probability (distribution) appropriate in any particular case under consideration. Thus its use is similar to that of "const" for an appropriate constant. The symbol "dens (. . ) " will, in the following, be used for "probability density" in a similar fashion. 6. The equations for Prob (x±%dx) are, of course, not rigorous: but they are so if the symbols " = " are interpreted as symbols of asymptotic equality.