Chapter 3 Random Variables - 3.4 Continuous Random Variables - Questions - Page 136: 4

$\color{blue}{\dfrac{26}{27}}$

$\begin{align*} P( Y\gt 1) &= \int_1^\infty f_Y(y)\ dy \\ &= \int_1^3 \frac{1}{9}y^2\ dy + \int_3^\infty 0\ dx \\ &= \left[ \frac{1}{9}\frac{y^3}{3}\right]_1^3 + 0 \\ &= \left[ \frac{1}{27}y^3\right]_1^3 \\ &= \frac{1}{27}(3^3-1^3) \\ &= \frac{1}{27}(27-1) \\ \color{blue}{P(Y\gt 1)} &\color{blue}{= \frac{26}{27}} \end{align*}$