Largest Number Drawn - RV From a box containing N identical tickets numbered 1 to N, n tickets are drawn with replacement. Let X be the largest number drawn. Find E[X]. Rohatgi 3.2 (2nd problem)
My solution 2:
I thought about using the Indicator RV.
Xi = 1, if the ith draw is the largest
Xi = 0, o/w
Now, E[Xi] = Pr {Xi = 1}
                      = 1/i {Since each of other i-1 numbers could be the largest}
=> E[X] = $\displaystyle\sum\limits_{i=1}^N 1/i$
Does it make sense?
Thanks.

Pick your $n$ numbers.  There are ${N^n}$ ways to do that.
The number of ways the maximum is $k$ is ${k^n-(k-1)^n}$.
The sum you want is $\sum_{k=1}^N k \Pr(\max=k)\\=
\sum_{k=1}^N [k^{n+1}-k(k-1)^n]/N^n\\=N-\sum_{k=1}^{N-1}(k/N)^n$
