Question

Suppose that two electronic components in the guidance system for a missile operate independently and that each has a life length governed by the exponential distribution with mean 1 (with measurements in hundreds of hours). 1 + a2Y2, then it can be shown that the If Y1 and Y2 are independent random variables with moment-generating functions my, (t) and my (t), respectively, and a, and an are constants, and U = a moment-generating function for U is my(t) = my, (at)my, (azt). Use the method of moment-generating functions to obtain the density function for U, the average life length of the two components. u20 0 Need Help? Read It Talk to a Tutor Submit Answer

Answer 1

Solution ilety, Nexp(meuns Ry 4₂ dae Independent, pdf ofy, fry=ė T. yzo we know mgf of 4 MICE) = (1-to-1 Let u=ay, + d2 Yz Mgf of u Mult ) = Erevt) - Elecy, theydt) = Eleantech let) = Ered (ast)) E (ez Co2t)) s indeportat. / Mult) = my, cast] My Cart) T Let u lit Myfofo Mult) = Eleut] - F (eth It ] = Elette 2 = Elett Elekter) = My, ( 1 ) M ( ) = (t f'(1 t) Mult) = (1 + 2 - mgf of Gamma (2, 2) Hence by uniqueness then u=1th Ngamma (212) pdf of v fulu) =sku e zu uso I uso