Take the tour ×  Possible Duplicate: Finding $E(N)$ in this
question  suppose $X_1,X_2,\ldots$ is sequence of independent
random variables of $U(0,1)$ if $N=\min\{n>0
:X_{(n:n)}-X_{(1:n)}>\alpha , 0<\alpha<1\}$ that $X_{(1:n)}$ is
smallest order statistic and $X_{(n:n)}$ is largest order
statistic. how can show $P(N>n)=P(X_{(n:n)}-X_{(1:n)}<\alpha)$
share|improve this question Actually, the question asked here
was also asked by the OP in a follow-up comment on Didier's
answer to the earlier question, and has been answered there by
Didier. –  Dilip Sarwate Aug 1 '12 at 14:09 add comment  marked
as duplicate by Dilip Sarwate, Did, t.b., Sasha, Ｊ. Ｍ. Aug 2 '12
at 5:14   1 Answer  up vote 1 down vote accepted  If
$(Z_n)_{n\geqslant0}$ is a sequence of random variables such
that $Z_n\leqslant Z_{n+1}$ for every $n\geqslant0$, then
$N_a=\inf\{n\geqslant0\,;\,Z_n\gt a\}$ is such that, for every
$n\geqslant0$, $$ [N_a\gt n]=[Z_1\leqslant a,\ldots,Z_n\leqslant
a]=[Z_n\leqslant a]. $$ Note: No probability here, this a