= Solution
Let $z_u,z_d$ be the state-price-density values at the higher and lower risky payoff. Pricing the two assets gives
$$
\tfrac12\,6(z_u+z_d)=5,\qquad\tfrac12(7z_u+4z_d)=5.
$$
Solving yields
$$
\boxed{Z=\begin{cases}10/9&S_1=7,\\5/9&S_1=4.\end{cases}}
$$
Both values are positive, and the two independent equations make the solution unique. The payoff matrix has nonzero determinant, so this is a <complete two-state market>. The actual <Arrow state prices>, including physical probabilities, are $q_u=5/9$ and $q_d=5/18$. Their sum is $5/6$, the <discount factor>. Normalizing gives <risk-neutral probabilities> $2/3,1/3$. Thus $Z$ itself is not a probability density of mean one: its mean is $5/6$ because the riskless asset earns interest.
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