10 ties in a wardrobe I have $10$ ties in a wardrobe and I pick one at random with probability $0.1$ each day independently. I return the tie after usage, so we allow replacement.
let $\,\theta$ be equal to the number of ties I used in course of $5$ days. what is the expected value and variance of $\, \theta$ random variable?
Now , I know this is some type of  application of indicators, I would like to know which theorem or method in regards to the indicators does this problem use for the solution?
 A: Here's the theory.
Suppose we let
$$X_i=
\begin{cases}
1 \quad \text{if tie i is used at least once in 5 days} \\
0 \quad \text{otherwise}
\end{cases}$$
for $i=1,2,3,\dots,10$. Then we will have
$$\theta = \sum_{i=1}^{10} X_i$$
so by linearity of expectation
$$E(\theta) = \sum_{i=1}^{10} E(X_i)$$
Note: The theorem that $E(X+Y)=E(X)+E(Y)$ does not require that $X$ and $Y$ be independent.  That's good for us here, because the $X_i$'s are not independent.  So the problem of finding the expected value of $\theta$ reduces to finding the expected value of $X_i$, which seems simpler, and since $X_i$ is either zero or one, we have $E(X_i) = P(X_i=1)$.
Finding the variance of $\theta$ is a little more complicated, and to apply the method of indicators we will need to find $E(X_i X_j)$ for $i < j$.  (Don't make the mistake of thinking $E(X_i X_j) = E(X_i)\cdot E(X_j)$ here!)  Assuming we can find $E(X_i X_j)$, recall that 
$$var(\theta) = E(\theta^2) - E(\theta)^2$$
We already know how to find $E(\theta)$, so it remains to find $E(\theta^2)$.  To do so, we apply the identity
$$ \left( \sum_{n=1}^{10} X_i \right)  ^2 = \sum_{n=1}^{10} X_i^2 + 2 \sum_{i <j} X_i X_j$$
In our case $X_i$ is always either zero or one, so $X_i^2 = X_i$, hence
$$\left( \sum_{n=1}^{10} X_i \right)^2 = \sum_{n=1}^{10} X_i + 2 \sum_{i <j} X_i X_j$$
Again applying linearity of expectation, 
$$E(\theta^2)=E \left[ \left( \sum_{n=1}^{10} X_i \right)^2 \right] = \sum_{n=1}^{10} E(X_i) + 2 \sum_{i <j} E(X_i X_j)$$
We assume we know how to find all the quantities on the right-hand side of the above equation, so we can find $E(\theta^2)$, which is the last step we need in finding $var(\theta)$.
