I am a little confused on how to go about finding different parts of the Variance of a random variable. Here is the question.
A total of $n$ balls, numbered $1,.. n$, are put into $n$ urns, also numbered $1,.. n$ in such a way that ball $i$ is equally likely to go into any of the urns $1,2,...,i$. Find the variance of of the number of empty urns.
This is what I have so far:
Let $X_i=1$ if empty, 0 otherwise. So $E(X_i)=1*P(X_i=1)+0*P(X_i=0)=P(X_i=1)$.
$Var(X)=(\Sigma^n_{i=1} Var(X_i))=\Sigma^n_{i=1}Var(X_i)+2\Sigma\Sigma_{i<j}Cov(X_i,X_j)$.
I know that the $Var(X_i)=E(X_i^2)-E^2(X_i)$ and $Cov(X_i,X_j)=E(X_iX_j)-E(X_i)E(X_j)$.
I think the main problem is I dont know what $P(X_i=1)$ is.