Science:Math Exam Resources/Courses/MATH101 B/April 2025/Question 03 (a)/Solution 1

To compute the probability that $X\le \frac{1}{2}$ we integrate from $0$ to $\frac{1}{2}$, $$\begin{array}{r}\underset{0}{\overset{\frac{1}{2}}{\int }}f(x)dx=\underset{0}{\overset{\frac{1}{2}}{\int }}2xdx=x^2{|}_0^{1/2}=\frac{1}{4}=25\mathrm{\%}.\end{array}$$ Alternatively, one could sketch $y=2x$ and identify the area under the line from $x=0$ to $x=\frac{1}{2}$ as a triangle of base $\frac{1}{2}$ and height $1$, and so an area of $$\begin{array}{r}\frac{bh}{2}=\frac{1/2\cdot 1}{2}=\frac{1}{4}=25\mathrm{\%}.\end{array}$$