Summation Simplification for Neumerator of Beta Estimator

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The equation Ʃwixiyi - (ƩxiwiƩyiwi)/Ʃwi can be simplified into the form Ʃ(something - something)yi, where the first "something" is indeed xiwi. The second "something" is the mean of x, represented as (Ʃxjwj/Ʃwj). Using separate indices for clarity helps in understanding the transformation. This simplification clarifies the relationship between the weighted sums and the mean. The discussion highlights the importance of clear notation in mathematical expressions.
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I need simplify this equation:

Ʃwixiyi - (ƩxiwiƩyiwi)/Ʃwi

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

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

Any help would be greatly appreciated. Thanks!
 
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LBJking123 said:
I need simplify this equation:

Ʃwixiyi - (ƩxiwiƩyiwi)/Ʃwi

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

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

Any help would be greatly appreciated. Thanks!

Come on, that's easy:
<br /> \sum_i w_i(x_i-\sum_j x_j w_j/\sum_k w_k) y_i<br />
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.
 
Ohhhhh I kept thinking it was going to be

Ʃ(xiwi-(xiwi2)/Ʃwi)yi

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

Thanks for the help DrDu!
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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