Suppose that $\displaystyle X_1,X_2, \ldots , X_n $ are i.d.d. to some distributions $\displaystyle ( \mu _X , \sigma ^2 _X ) $ and $\displaystyle Y_1, \ldots , Y_m $ are i.d.d. to $\displaystyle ( \mu _Y , \sigma _Y ^2 ) $.

If all the X's and Y's are indepedent to one aother, does that also mean that the sums $\displaystyle X_1 + \ldots + X_n $ is also independent to $\displaystyle Y_1 + \ldots + Y_m $?

So I'm trying to prove it by doing it striaght from the definition:

$\displaystyle Pr[ X_1 + \dots + X_n \cap Y_1 + \dots + Y_m ] $ and try to get it to become $\displaystyle Pr[X_1 + \dots + X_n ] \cdot Pr[Y_1 + \dots + Y_m ] $

But I'm getting stuck on the left hand side, not sure how to break it down...

Any help, please? Thank you!