  For a 1-parametric family $E_k$ of elliptic curves over $\mathbb{Q}$ and a
prime $p$, consider the second moment sum $M_{2,p}(E_k)=\sum_{k \in
\mathbb{F}_p} a_{k,p}^2$, where $a_{k,p}=p+1-\#E_k(\mathbb{F}_p)$. Inspired by
Rosen and Silverman's proof of Nagao conjecture which relates the first moment
of a rational elliptic surface to the rank of Mordell-Weil group of the
corresponding elliptic curve, S. J. Miller initiated the study of the
asymptotic expansion of $M_{2,p}(E_k)=p^2+O(p^{3/2})$ (which by the work of
Deligne and Michel has cohomological interpretation). He conjectured, in
parallel to the first moment case, that the largest lower order term that does
not average to 0 is on the average negative (i.e. has a negative bias).
  Miller verified the Bias Conjecture for the number of pencils of cubics
$E_k:y^2=P(x)k+Q(x)$, where $\textrm{deg} P(x), \textrm{deg} Q(x) \le 3$.
  In this paper, assuming Sato-Tate conjecture for genus two curves, we prove
that the Bias Conjecture holds for the pencil of the cubics E_k with the
generic choice of polynomials $P(x)$ and $Q(x)$ with $\textrm{deg} P(x),
\textrm{deg} Q(x)\le 3$.
