jmc

Each outcome of rolling an octahedral (8-sided) die has probability $\frac{1}{8}$, and the possible outcomes are 1, 2, 3, 4, 5, 6, 7, and 8. So the expected value is $$\frac{1}{8}(1)+\frac{1}{8}(2)+\frac{1}{8}(3)+\frac{1}{8}(4)+\frac{1}{8}(5)+\frac{1}{8}(6)+\frac{1}{8}(7)+\frac{1}{8}(8)=\frac{36}{8}=\boxed{4.5}.$$