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November 17th, 2017, 01:30 PM   #1
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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.
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November 17th, 2017, 09:44 PM   #2
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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 $(n-1)!$ orderings where $Y_m$ is the max.

There are a total of $n!$ orderings.

$P[Y_m \text{ is max of }n] = \dfrac{(n-1)!}{n!} = \dfrac 1 n$
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