Summation Simplification for Neumerator of Beta Estimator

[LBJking123]

I need simplify this equation:

Ʃw_ix_iy_i - (Ʃx_iw_iƩy_iw_i)/Ʃw_i

Into an equation of the form: Ʃ(something - something)y_i

I am pretty sure the first something is x_iw_i, but I have no idea what the second something would be...

Any help would be greatly appreciated. Thanks!

 

[DrDu]

I need simplify this equation:

Ʃw_ix_iy_i - (Ʃx_iw_iƩy_iw_i)/Ʃw_i

Into an equation of the form: Ʃ(something - something)y_i

I am pretty sure the first something is x_iw_i, but I have no idea what the second something would be...

Any help would be greatly appreciated. Thanks!

Come on, that's easy:
$\sum_i w_i(x_i-\sum_j x_j w_j/\sum_k w_k) y_i$
Note that I chose a separate index for each summation for clarity.
The term in brakets is x minus <x>, i.e. the mean of x.

 

[LBJking123]

Ohhhhh I kept thinking it was going to be

Ʃ(x_iw_i-(x_iw_i²)/Ʃw_i)y_i

I makes more sense when you choose a separate index for each summation.

Thanks for the help DrDu!