Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 17 (c)/Solution 1

$${\begin{aligned}\mathbb {E} (X)&=\int _{-\infty }^{\infty }xf(x)\;\mathrm {d} x=\int _{-1/2}^{0}x\;\mathrm {d} x+\int _{0}^{1/4}2x\;\mathrm {d} x\\&=\left.\left[{\frac {x^{2}}{2}}\right]\right|_{-1/2}^{0}+\left.\left[x^{2}\right]\right|_{0}^{1/4}=0-{\frac {1}{8}}+{\frac {1}{16}}-0=-{\frac {1}{16}}.\end{aligned}}$$