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November 17th, 2017, 02:30 PM  #1 
Member Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1  Probability
Suppose A$_1$,A$_2$,... are independent random variables with the uniform distribution in [0,1]. We say that n is a record if: A$_n$ $\geq$ A$_m$ for all 1 $\leq m \leq$ n 1.Show that the probability of the event Y$_n$ that n is a record is P[Y$_n$]= 1/n 2.Let Z denote the event that there are infinitely many records. Show that P [Z] is zero or one. 
November 17th, 2017, 10:44 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,324 Thanks: 1233 
for (1) As all the $Y_k$'s are identical all the possible orderings of them are equiprobable. For a given $Y_m$ there are $(n1)!$ orderings where $Y_m$ is the max. There are a total of $n!$ orderings. $P[Y_m \text{ is max of }n] = \dfrac{(n1)!}{n!} = \dfrac 1 n$ 

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probability, random variable, record 
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