Suppose the following situation:
- We have $n$ marbles and these are tossed so that each marble lands in one of the $k$ boxes.
- The probability that a marble lands in box $\#i$ is $p_{i}\,,\ i=1,...,k\,,\ \sum_{i = 1}^{k}p_{i} = 1$.
- Then let $X_{l}$ be the number of marbles landing in box $\#l\,,\ l = i,j$. Define $Y_{l} = 1$ if $l^{th}$ marble lands in box $\#i,0$ otherwise. Similarly $Z_{l} = 1$ if $l^{th}$ marble lands in box $\#j,0$ otherwise.Here $i\neq j$. Also, write $X_i = \sum_{l = 1}^{n}Y_{l}$ and $X_{j} = \sum_{l = 1}^{n}Z_{l}$.
- Now follows a computation of $\mathrm{Cov}\left(X_{i},X_{j}\right)$ which ends with $=n\left \{E\left[Y_{1}Z_{1}\right]-E\left[Y_{1}\right]E\left[Z_{1}\right]\right\}$ and they claim:
"It is not hard to see that $E\left[Y_{1}Z_{1}\right] = 0$".
I do not understand here why ?. To see the source type
"It is not hard to see that E" inference nitis
into Google.