We consider a finite system $F={f_1,...,f_m}$ of maps defined on a compact interval or on a half-line. Assume that all elements of $F$ are strictly increasing and convex (but not necessarily strictly convex). We do not assume that all of them are contractive. There may even exist a common repelling fixed point. The system $F$ together with a probability vector $(p_1,...,p_m)$ forms a random iterated function system (RIFS). We apply $f_i$ at each step independently with probability $p_i$. Under the condition that the RIFS is contracting on average in a rather broad sense, we prove that the Hausdorff dimension of any invariant measure (there may be more than one) is less than or equal to the entropy divided by the Lyapunov exponent. This is joint work with Ai-Hua Fan and H. Toth.