Functions of One Random Variable

Solution 1:

For $(b)$, if you are first finding CDF and then taking its derivative,

$F_Y(y) = P (Y \lt y) = P( - \sqrt y \lt X \lt \sqrt y) $

$i$) If $ ~- \sqrt y \gt -1$ and $\sqrt y \lt 1~$, then

$ \displaystyle F_Y(y) = \int_{-\sqrt y}^{\sqrt y} \frac 2 9 (x+2) ~ dx = \frac {8\sqrt y}{9}$

So, $ \displaystyle f_Y(y) = \frac{4}{9\sqrt y}, 0 \lt y \lt 1$

$ii$) If $ ~- \sqrt y \lt - 1$,

$ \displaystyle F_Y(y) = \int_{-\sqrt y}^{-1} \frac 29 (x+2) ~ dx = \frac{4 \sqrt y - y - 3}{9}$

Or, $ \displaystyle f_Y(y) = \frac{2 - \sqrt y}{9\sqrt y}, ~1 \lt y \lt 4$