By William Feller
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Additional info for An Introduction to Probability Theory and Its Applications, Vol. 2
L V(tfx) dx t > O. 4) f. oo v(y) -dy , y t , t> O. v(t) = -tf'(t), We have thus found the analytic relationship between the density v of the length of a random vector in 3',3, and the density f of the length of its projection on a fixed direction. 4) in the opposite direction. ) Examples. (d) Maxwell distribution for velocities. Consider random vectors in space whose projections on the' x-axis have the normal density with zero expectation and unit yariance. 5) f(t) = 2n(t) = J2/7T e-1tl , . t > O.
Random vectors in 31 2 are defined in like manner. The distribution V of the true iength and the distribution F of the projection are related. 8) . '1f o . )dO. F we must depend on the relatively deep theory of Abel's inte~1 equation. 1, We state without proof that if F has a continuous density f, then. 1 ,.... 9) f( . d~ 0. ) Example. (j) Binary orbits. In obserVing a spectroscopic binary orbit astronomers can measure only the projections of vectors onto a plane perpendicular to the line of Sight.
The sample' space corresponding to two independent variables ,X and Y' that. , and probabilities in it are defined by their area. The same idea applies to triple,s and n·,tuples. " The result of theconcepttial experiment "n independent random choices of a point in 0, 1" requi~es an n-dimensional hypercube for its probabilistic description, but the experiment as such yields n points X~, .. ; ,Xn in the same interval. With unit probability no two of them are equal, and hence they partition 0, 1 into n + 1 subintervals.