For a sequence ${{eta}_m}_m$ of unit vectors in $mathbb{C}^n$, we consider the associated linear functional ${omega}$ on the Cuntz algebra $mathcal{O}_n$. We show that the restriction ${omega}{mid}_{UHF_n}$ is the product pure state of a subalgebra $UHF_n$ of $mathcal{O}_n$ such that ${omega}{mid}_{UHF_n}={otimes}{omega}_m$ with ${omega}_m({cdot})=<;{cdot}{eta}_m,{eta}_m;>$. We study product pure states of UHF and obtain a concrete description of them in terms of unit vectors. We also study states of $UHF_n$ which is the restriction of the linear functionals on $O_n$ associated to a fixed unit vector in $mathbb{C}^n$.