Finding Expected Value of vector length? We build a vector of digits, such that every time we pick a number randomly from {0,1,2,3,...,9} (For each digit the possibility of picking it is equal this 1/10 and not related to previous choices). We stop building out vector at the first time each digit appears at least once (Which means the last digit appears exactly once or else we can have a shorter vector which is a contradiction) Now I want to calculate Expected value of length of the vector created.
How can I solve this, I tried with indicators but don't think of any specific indicator what will be helpful here.
 A: Let's consider a similar problem.
Suppose I had a set of $p$ elements, and a subset of $q\leq p$ elements. What is the expected number of draws with replacement from the set of $p$ elements before we get one of the $q$ elements?
The answer is $\frac pq$.
So, our answer is $$\frac{10}1+\frac{10}2+\ldots+\frac{10}{10}=\frac{7381}{252}$$
A: We can split up the procedure into several waiting times.
Let $T_1$ be the number of draws before seeing the first new number. We have $T_1=1$ since the first draw will give you the first new number.
Let $T_2$ be the number of subsequent draws before seeing the second new number. So for instance, if your sequence is $3,3,3,3,1$, then $T_2=4$ since your first new number is "3", and you had to draw $3,3,3,1$ before seeing the second new number "1".
Let $T_3$ be the number of subsequent draws before seeing the third new number, and so on.
In the end, the total number of draws is $T_1 + T_2 + \cdots + T_{10}$. By linearity of expectation,
$$E[T_1 + T_2 + \cdots + T_{10}] = E[T_1] + E[T_2] + \cdots + E[T_{10}],$$
so it suffices to compute each expectation separately.
Recognize that $T_i$ is a geometric distribution. In a sequence of independent draws of successes (seeing the $i$th new number) and failures (seeing an old number), it counts the number of draws until the first success. Note that a success her happens with probability $1 - \frac{i-1}{n}$ since there are $i-1$ "old" numbers you have already seen so far. You can then refer to your prior knowledge about the expectation of a geometric random variable to compute $E[T_i]$.
