What are the functional form of the fixed-effects regression, random-effects regression and pooled regression?

[Lucas Bordure]

Good morning to the community Stata.

I wrote that the functional form of the fixed-effects regression was:
Y_it=α_i+ 〖X'〗_it β+ε_it

I wrote that the functional form of the random-effects regression was:
Y_it=〖X'〗_it β+( α_i+v_(it ) )

I wrote that the functional form of the pooled regression was:
Y_it=β_0+ 〖X'〗_it β+u_it

Is that correct?