142 Let Y1 Y2 be independent and identically distributed wi

1.42. Let Y1, Y2,. . be independent and identically distributed with P{Yn = 0}= alpha P{Yn > y} = (1 - alpha)e^- y > 0. REFERENCES Define the random variables Xn, n = > 0 by X0 = 0 Xn+1 = aXn + Yn+1 Prove that P{Xn= 0} = alpha^n P{Xn > x} = (1 - alpha^n)e^-x, x > 0

Solution

x0 =0

x1 =ax0+y1

x2= ax1+y2 = a(ax0+y1)+y1+y2

= a²x0+ay1+y2

Similarly

xn = aⁿx0+a^n-1y1+...+yn

Hence P(Xn=0) = P(aⁿx0+a^n-1y1+...+yn=0) = aⁿ

P(xn>x) = 1-P(X<x) = (1-aⁿ)e^-x