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Let $X_1, X_2, \ldots$ be independent and identically distributed continuous random variables. Let $N$ be the smallest value of $n$ for which $X_n > X_1$. Show that $P(N > k) = 1/k$ (for $k = 1, 2, \ldots$) and hence that $P(N = k) = 1/[k(k-1)]$. What is $E(N)$?

I have no idea where to start, any hint/sketch of the solution would be much appreciated. Thank you!

Does E(N) exist, since using the definition gives the harmonic series which diverges

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  • $\begingroup$ For $k$ iid random variables, the probability that the first one is greater than the others is $1/k$, by symmetry. $\endgroup$ – Paul Feb 8 '17 at 22:20
  • $\begingroup$ But how do I prove this? $\endgroup$ – Romeo123 Feb 8 '17 at 22:22
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As Paul commented, you can use symmetry.

If $N>k,$ this means that $X_1$ is the largest of the first $k$ random variables. So $$ P(N>k) = P(X_1>X_2, X_1>X_3,\ldots X_1>X_k) $$ By symmetry, each ordering of the random variables is equally probable. There are $k!$ total orderings and $(k-1)!$ orderings in which $X_1$ is largest. Thus $$P(N>k) = \frac{(k-1)!}{k!} = \frac{1}{k}.$$ Or you can just say each of the $k$ variables is equally likely to be largest, and $\sum_{i=1}^kP(\mbox{$X_i$ is largest})=1$ so $P(\mbox{$X_1$ is largest}) = 1/k.$

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  • $\begingroup$ Thank you? How do I do E(N)?? $\endgroup$ – Romeo123 Feb 8 '17 at 22:41
  • $\begingroup$ @Romeo123 Well, you know $E(N) = \sum_k k P(N=k).$ Can you figure out $P(N=k)$ from knowing $P(N> i )$ for all $i$? $\endgroup$ – spaceisdarkgreen Feb 8 '17 at 22:55
  • $\begingroup$ @Romeo123 Alternatively, there's an expression out there for $E(N)$ in terms of $P(N>k)$ which you can derive from the identity $N =1+ \sum_{k=1}^\infty I(N > k)$ where $I$ is the indicator function (after convincing yourself the identity is true). $\endgroup$ – spaceisdarkgreen Feb 8 '17 at 23:02
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It might be intuitive from the symmetry perspective. A rigorous argument is as follows:

Let $F$ be the distribution function of $X_1, X_2, \ldots$, by the definition of the random variable $N$ and the independence assumption, it follows that \begin{align} & P[N > k] = P[X_2 \leq X_1, X_3 \leq X_1, \ldots, X_k \leq X_1] \\ = & \int_{-\infty}^\infty P[X_2 \leq x, \ldots, X_k \leq x] dF(x) \\ = & \int_{-\infty}^\infty P[X_2 \leq x]\cdots P[X_k \leq x] dF(x) \\ = & \int_{-\infty}^\infty F(x)^{k - 1} d F(x) = \int_0^1 u^{k - 1} du \\ = & \frac{1}{k}. \end{align}

Here in the last step, we need the condition that $F$ is continuous so that the change of variable works.

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  • $\begingroup$ One should probably add that all random variables have a density function (since this is not true for general random vectors et cetera) $\endgroup$ – Adam Hughes Feb 8 '17 at 22:54
  • $\begingroup$ @AdamHughes I think a minimal condition would be $F$ is continuous. For which case we can show (for a proof, see this post) that $F(X)$ has uniform distribution on $[0, 1]$. The existence of density might be too strong. $\endgroup$ – Zhanxiong Feb 8 '17 at 23:23
  • $\begingroup$ how does one define $dF$ if not as a density? As a second factor I'm not sure I see continuity as a weaker condition as a contiguous CDF implies absolute continuity wrt Lebesgue measure, hence existence of the Radon-Nikodym derivative, so the implication is the other direction. $\endgroup$ – Adam Hughes Feb 8 '17 at 23:35
  • $\begingroup$ The notation $dF(x)$ is a shorthand for $F(dx)$, or $\mu(dx)$, where $\mu$ is the measure on $\mathbb{R}^1$ induced by $F$. To use this notation, we do not require $\mu$ has a density $f$ with respect to the Lebesgue measure. $\endgroup$ – Zhanxiong Feb 8 '17 at 23:42
  • $\begingroup$ I absolutely agree. In fact, as I think about this, I'm pretty sure that this is equivalent to density's existence. The real issue I think is that the reason you can integrate from 0 to 1 without skipping anything is because of continuity, the change of integrand is density existence. $\endgroup$ – Adam Hughes Feb 8 '17 at 23:47

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