Question

Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.

f(x) = ${\displaystyle \{\begin{array}{ll}0.5x& \text{for}0\le x\le 2\\ 0& \text{otherwise}.\end{array}}$
Calculate : P(0.5 ≤ X ≤ 1.5)

Answer 1

P(0.5 ≤ X ≤ 1.5) = ${\displaystyle {\int }_{0.5}^{1.5}f\left(x\right)\cdot dx}$

= ${\displaystyle 0.5{\int }_{0.5}^{1.5}x\cdot dx}$

= ${\displaystyle \frac{0.5}{2}{\left[x^2\right]}_{0.5}^{1.5}}$

= ${\displaystyle \frac{1}{4}[2.25-0.25]}$

= ${\displaystyle \frac{1}{4}\times 2}$

= ${\displaystyle \frac{1}{2}}$.