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Homework Help: Sequence of random variables

  1. Mar 31, 2015 #1
    Hi, I'm trying to solve this exercise but I really don't know how
    1. The problem statement, all variables and given/known data

    Let X1, X2,.. be a sequence of iid random variables following a uniform distribution on (0,1). Define the random variable N≥2 as the first point in which the sequence (X1,X2,...) stops decreasing. i.e If N=n :
    X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifX2[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gif...[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifXn-1<Xn [Broken]

    For 0[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gif1 [Broken] show that

    1. P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N=n)=tn-1/(n-1)! [Broken] - tn/(n)!
    2.P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N [Broken] pair)=1-exp(-t) use series expansion of exp(t)+exp(-t) et exp(t)-exp(-t)
    3. E[N]=e

    2. Relevant equations

    3. The attempt at a solution
    I solved the third one
    For the first one, I think that we have to use induction proof, but I don't see how to do that here ?
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Mar 31, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    For the first one, try out some simple cases first: do it for n = 2, n = 3, etc. You will soon see how to do the general case.

    What you wrote for 1) is wrong: you say
    [tex] P(X_1 \leq t, N=n) = tn - \frac{1}{(n-1)!} - \frac{tn}{n!}[/tex]
    when your expression is parsed using standard mathematical rules. I suspect you might have meant
    [tex] \frac{t^{n-1}}{(n-1)!} - \frac{t^n}{n!} [/tex]
    In that case, you MUST use "^" signs and parentheses: tn means ##t \times n##, but t^n means ##t^n##. Similarly, t^n-1 means ##t^n - 1##, but t^(n-1) means ##t^{n-1}##.

    Finally, I do not understand what "X1 ≤ t, N pair" means---that is, what is "N pair"?
    Last edited by a moderator: May 7, 2017
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