zz-28 - Utah State University ECE 6010 Stochastic Processes...

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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...
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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.

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zz-28 - Utah State University ECE 6010 Stochastic Processes...

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