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Uniform Distribution over n and its limit

  1. Jan 28, 2013 #1
    1. The problem statement, all variables and given/known data

    Let Yn be uniform on {1, 2, . . . , n} (i.e. taking each value with probability 1/n). Draw the distribution function of Yn/n. Show that the sequence Yn/n converges in distribution as n → ∞. What is the limit?

    2. Relevant equations

    So Yn has c.d.f Yn(x) = |x|/n where |x| is the nearest integer less than x.

    3. The attempt at a solution

    Is it ok for me to just divide the c.d.f of Yn by n, and so plot |x|/n2?
    This seems far too easy for 2nd year of university (I really do not trust myself with probability) and also it only gives a total for x > n of 1/n. Is this allowed?

    If so, then Yn/n clearly tends to 0 as n → 0 since we have 1/n.

    Thanks
     
    Last edited: Jan 28, 2013
  2. jcsd
  3. Jan 28, 2013 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Your notation is horrible and will get you into trouble. Yn is a random variable, not a function of x. Its distribution function is [tex] F_n(x) = P(Y_n \leq x) = \left\{
    \begin{array}{cl}\frac{\lfloor x \rfloor}{n},& \: 0 \leq x < n\\
    1,&\: x \geq n
    \end{array}\right. [/tex] as you said. The distribution of Y_n/n is
    [tex] G_n(x) = P\left( \frac{Y_n}{n} \leq x \right) = P( Y_n \leq n x ) = F_n(nx)
    = \left\{ \begin{array}{cl}
    \frac{\lfloor nx \rfloor}{n}, &\: 0 \leq nx < n\\
    1, & \: nx \geq n
    \end{array} \right. [/tex]
     
  4. Jan 29, 2013 #3
    Thank you for your clarification on notation and help!
     
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