This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Utah State University ECE 6010 Stochastic Processes Homework # 6 Solutions 1. Suppose { X n } ) ∞ n =1 is a sequence of independent r.v.s each of which is uniformly dis tributed on the interval (0 , 1). Define a sequence of r.v.s { Z n } by Z n = n (1 Y n ), where Y N = max 1 ≤ i ≤ n X i . Show that { Z n } ∞ n =1 converges in distribution to an exponential r.v. with p.d.f. f ( x ) = ( e x x ≥ otherwise . Here, F Z n ( z ) = P ( Z n ≤ z ) = P ( n (1 Y n ) ≤ z ) = P ( Y n ≥ (1 z/n )) = 1 P ( y n < (1 z/n )) = 1 P ( max 1 ≤ i ≤ n X i < (1 z/n )) = 1 P ( X 1 < (1 z/n ) , X 2 < (1 z/n ) , . . . , X n < (1 z/n )) = 1 P ( X 1 < (1 z/n )) · P ( X 2 < (1 z/n )) ··· P ( X n < (1 z/n )) Now, P ( X i < (1 z/n )) = (1 z/n ) < 0that is, if z > n 1 z/n ≤ (1 z/n ) ≤ 1that is, if 0 ≤ z ≤ n 1 (1 z/n ) > 1that is, if z < therefore, F Z n ( z ) = z > n 1 (1 z/n ) n ≤ z ≤ n 1 z < we have lim n →∞ (1 z/n ) n = e z , so...
View
Full
Document
This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.
 Spring '08
 Stites,M

Click to edit the document details