# Homework Help: Uniform Distribution over n and its limit

1. Jan 28, 2013

### JonoPUH

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. Jan 28, 2013

### Ray Vickson

Your notation is horrible and will get you into trouble. Yn is a random variable, not a function of x. Its distribution function is $$F_n(x) = P(Y_n \leq x) = \left\{ \begin{array}{cl}\frac{\lfloor x \rfloor}{n},& \: 0 \leq x < n\\ 1,&\: x \geq n \end{array}\right.$$ as you said. The distribution of Y_n/n is
$$G_n(x) = P\left( \frac{Y_n}{n} \leq x \right) = P( Y_n \leq n x ) = F_n(nx) = \left\{ \begin{array}{cl} \frac{\lfloor nx \rfloor}{n}, &\: 0 \leq nx < n\\ 1, & \: nx \geq n \end{array} \right.$$

3. Jan 29, 2013

### JonoPUH

Thank you for your clarification on notation and help!