Answer
$$\frac{3}{4} v_{\mathrm{o}}
$$
Work Step by Step
The average speed is
$$
v_{\text {avg }}=\int_{0}^{v_{0}} v P(v) d v=\int_{0}^{v_{0}} v\left(\frac{3 v^{2}}{v_{\mathrm{o}}^{3}}\right) d v=\frac{3}{v_{\mathrm{o}}^{3}} \int_{0}^{v_{0}} v^{3} d v=\frac{3}{4} v_{\mathrm{o}}
$$
