Using the definition of
mean 1 as a sum,
\begin{align*}
\mu & = 0 \cdot 0.10 + 1 \cdot 0.25 + 2 \cdot 0.40 + 4 \cdot 0.15 + 7 \cdot 0.10\\
& = 0 + 0.25 + 0.80 + 0.60 + 0.70\\
& = 2.35
\end{align*}
Notice where this lies on the probability histogram for this distribution.