Solution
= Solution
A <probability density function> must be nonnegative and have <definite integral> one. Here
$$
\int_{-\infty}^{\infty}f(x)\,dx=\alpha\int_0^2x^2\,dx=\frac{8\alpha}{3}.
$$
Therefore
$$
\boxed{\alpha=\frac38,\qquad f\text{ is a valid density}.}
$$
This value is positive, so the nonnegativity requirement also holds.