\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider the function \(f(x) = a \sqrt{x}\).
Find the value of \(a\) such that \(f(x)\) is a probability density function on the interval \([0, 4]\).
\FieldText{1}{$$\begin{aligned}[t] 1&= \int_{0}^{4} a \sqrt{x} \, dx \\1&= a \left[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right]_{0}^{4} \\1&= a \cdot \frac{2}{3} \left( 4^{\frac{3}{2}} - 0 \right) \\1&= a \cdot \frac{2}{3} \cdot 8 \\1&= a \cdot \frac{16}{3} \\a &= \frac{3}{16}\end{aligned}$$}