Random vectors inducing absolutely summing operators

It will be proved that given a random vector X in a Banach space F such that $E(x*(X)^2)^ \ c (E|x*(X)|)^2 $ for some $c$ and all $x* \in F*$ then the operator $T*:F* \goes L_2(\Omega)$ given by $T*x* = x*(X)$ is 1- absolutely summing.