Course in Probability Theory

6.1 GENERAL PROPERTIES ; CONVOLUTIONS 1 1 53PROOF .We wish to show that(4) Vx : ,ql{Xi + X2 < x) = (F 1 * F2)(x) .For this purpose we define a function f of (x1, x2 ) as follows, for fixed x :_ 1, if x1 + x2 < x ;f (x1, x2) 0, otherwise .f is a Borel measurable function of two variables . By Theorem 3 .3 .3 andusing the notation of the second proof of Theorem 3 .3 .3, we havef f(X1, X2)d?T = fff(xi,x2)92µ 2 (dxi,dx2)= f 112 (dx2) f (XI, x2µl (dxl )= f 1L2(dx2) f µl (dxl )(-oo,x- X21= f00dF2(x2)F1(x - X2)-This reduces to (4) . The second equation above, evaluating the double integralby an iterated one, is an application of Fubini's theorem (see Sec . 3 .3) .Corollary . The binary operation of convolution * is commutative and associative.For the corresponding binary operation of addition of independent r .v .'shas these two properties .DEFINITION . The convolution of two probability density functions P1 andP2 is defined to be the probability density function p such that(5)and written asdx E R,' :P(x) ='COOP1(x - Y)P2(), ) dy,P=P1*P2 •We leave it to the reader to verify that p is indeed a density, but we willspell out the following connection .Theorem 6 .1 .2. The convolution of two absolutely continuous d .f .'s withdensities p1 and P2 is absolutely continuous with density P1 * P2 .