Consider the birthday of $98$ people on different days of the week. Assume that the birthdays of these people are independent and can be on any day of the week with the same probability. Find the covariance of the number of people born on Sunday and the number of people born on Monday.
I personally think the following is an answer but I am not sure about my idea:
Assume that $I_i (1\leq i\leq 98)$ is the indicator function such that $I_i=1$ if the ith person has born on Sunday and similarly assume that $S_i (1\leq i\leq 98)$ is the indicator function such that $S_i=1$ if the ith person has born on Monday. So we have to calculate the following:
$Cov(\sum_{j=1}^{98} I_j,\sum_{k=1}^{98} S_k)$
so we have :
$Cov(\sum_{j=1}^{98} I_j,\sum_{k=1}^{98} S_k) = \sum_{j=1}^{98}\sum_{k=1}^{98} Cov(I_j,S_k) $
and because of the fact that the birthdays are independent, this sum is equal to zero.
Am I calculating this correctly? If not, what is the right answer?
$I_j$ and $S_k$ are independent when $i\neq j$ but $I_j$ and $S_j$ are not. You should have $$\sum_{k=1}^{98}Cov(I_k,S_k)=98Cov(I_1,S_1)$$