Question

Escalate - Show every mynute steps in DETAILED! Explain & SHOW how the LIMITS of integration & DOMAIN are found.

Show, by using moment generating functions, that a random variable X, whose density function is e-l2\/2, X E R, can be written as X = Yı-Y2, where Y, and Y2 are independent exponentially distributed random variables.

Answer 1

*, *. ~E 1 ) . Y. . 70 fx, (Y)= 2*, 480 Map of X,: My CED= E(2+1). get e sacody = Fiat,c* day Parcosy e percen t w [147] Simce , Y, and Y have come distribution, Mx (+)= t; ltro MG F8f*: Mx (t) = E(etx)=E(et(71-727)Elety, 4E+SYz) = E(FY) E (CET)Y) + My, (+) Meyx[+] [:Y, and Yzare independent - onts contes into [it70,1-+70,-K +<1.7Itki]. Maff*: Mx (4) - Elety) - Iete fu code = retten van de - jetz Letalhes ſetzte talde ={;* 2,

jceveda - Castell di . thiyo (4-1)-PS -o o 1 + x=le-(int) 27 -(1-to do (1-7) [Note that of El-t) to then e-(1-t) 0 = 0º = 0o (not valid)] | Similasly in the above integral, (I++) >072(t+1)00.00 ] :: Mx Ct). [titi] : it. [ ++120, 1-t70, 141<1] By uniqueness of MGF, X=Y-Y2 where Y, and Y₂ ~ Exp (1) Note: Y,>o, Y220 3 0<x Loo ].