The expectation is the integral of $x$ times its probability density function. So, we have:
$$E(X) = \int_{0}^{1} x \cdot f(x) \, dx = \int_{0}^{1} x \cdot 20(x^{3}-x^{4}) \, dx$$
After calculating the integral, we find that $E(X) = \frac{2}{3}$.
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