\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider the function \(f(x) = a x^3\).
Find the value of \(a\) such that \(f(x)\) is a probability density function on the interval \([0, 2]\).\FieldText{1}{$$ \begin{aligned}[t]1&=\int_{0}^{2} f(x) \, dx &= \int_{0}^{2} a x^3 \, dx \\1&= a \int_{0}^{2} x^3 \, dx \\1&= a \left[ \frac{x^4}{4} \right]_{0}^{2} \\1&= a \left( \frac{2^4}{4} - \frac{0^4}{4} \right) \\1&= 4a \\a&=\frac{1}{4}\\\end{aligned}$$}