By Howard G. Tucker and Ralph P. Boas (Auth.)

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**Sample text**

X and Y are two bivariate) distribution function Fx,y( > y) is defined by Fx,Y(x,y) =P([I^P^]). Joint distribution functions have much the same properties as distri bution functions of one random variable (or, the so-called univariate dis tribution function). 4 that only hints of the proofs will be supplied. Formal proofs of these theorems are left to the student. 1. Theorem. then / / X and Y are random variables, if Xi S x2 and if y\ ^ y2, x Fx,y(xi, Hint of proof: yi) ^ FXl v( h 2/2). Use the fact that [X ^ xi][Y S yi] C [X ^ x2][Y ^ y2], 2.

The differential equation p£(t) = -\P (t) is easy to solve. We first rewrite it as P'0(t) + \PQ (t) = 0, Q Sec. 3] NOTION u O F INDEPENDENCE 25 then multiply both sides by e , and finally notice that what we have may be written as | ( i W ) = o. If we integrate both sides from 0 to t, weXobtain P0(t)e ( Po(0 = e- . , we solve +XP0(t). P[{t) = -\Pi(t) X( Since we now know that P0(t) = e~ , we may write P[{t) X l+ \Pi(t) = Xe-H Again, multiply both sides by e to obtain eX ft ™> ') X = - Xt Upon integrating both sides between 0 and t, one obtains Pi(t)e X£.

We first rewrite it as P'0(t) + \PQ (t) = 0, Q Sec. 3] NOTION u O F INDEPENDENCE 25 then multiply both sides by e , and finally notice that what we have may be written as | ( i W ) = o. If we integrate both sides from 0 to t, weXobtain P0(t)e ( Po(0 = e- . , we solve +XP0(t). P[{t) = -\Pi(t) X( Since we now know that P0(t) = e~ , we may write P[{t) X l+ \Pi(t) = Xe-H Again, multiply both sides by e to obtain eX ft ™> ') X = - Xt Upon integrating both sides between 0 and t, one obtains Pi(t)e X£.