1. ## Conditional covariance

$X_i\sim Ber(\frac{1}{4})$ and $Y_i\sim Ber(\frac{3}{4})$. Find $Cov(\sum_{i=1}^{N} X_i,\sum_{i=1}^{N} Y_i)$ where $P(N=k)=\frac{1}{2^k}$ for $k=1,2,3,....$

I thought using conditional expectation might be a good idea but got stuck.

2. Hello,

And what's the covariance between the Xi and Yi ?

In order not to be stuck, you may change the indices :

$cov\left(\sum_{i=1}^N X_i,\sum_{j=1}^N Y_j\right)=E_N\left[\cov\left(\sum_{i=1}^N X_i,\sum_{j=1}^N Y_j \mid N\right)\right]$
$=E\left[\sum_{i=1}^N\sum_{j=1}^N cov(X_i,Y_j)\right]$

For example, if $cov(X_i,Y_j)=a \quad \forall i,j$, then this will be equal to $a \cdot E[N^2]$

3. Yes; and here comes my problem: how will I calculate $E(XY)$? It gets complicated.

4. Originally Posted by Sambit
Yes; and here comes my problem: how will I calculate $E(XY)$? It gets complicated.
Because you're missing information !!!!
We don't even care about the distributions of X and Y, we care about their joint distribution (their covariance)

5. $(X,Y)$ jointly follows a multinomial distribution with parameters $N,\frac{1}{4},\frac{3}{4}$. So the covariance is $-N\frac{1}{4}\frac{3}{4}$