  We study approximation of multivariate functions from a separable Hilbert
space in the randomized setting with the error measured in the weighted $L_2$
norm. We consider algorithms that use standard information $\Lambda^{\rm std}$
consisting of function values or general linear information $\Lambda^{\rm all}$
consisting of arbitrary linear functionals. We use the weighted least squares
regression algorithm to obtain the upper estimates of the minimal randomized
error using $\Lambda^{\rm std}$. We investigate the equivalences of various
notions of algebraic and exponential tractability for $\Lambda^{\rm std}$ and
$\Lambda^{\rm all}$ for the normalized or absolute error criterion. We show
that in the randomized setting for the normalized or absolute error criterion,
the power of $\Lambda^{\rm std}$ is the same as that of $\Lambda^{\rm all}$ for
all notions of exponential and algebraic tractability without any condition.
Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak and
H.Wo\'zniakowski in the book: Tractability of Multivariate Problems, Volume
III: Standard Information for Operators, EMS Tracts in Mathematics, Z\"urich,
2012.
