Suppose that Y is a continuous random variable with a density function $f_{Y}(y)$. We transform $Y$ by the following mapping
\begin{equation} Y^{*} = \left \{ \begin{array}{ll} \alpha Y + \beta & \text{ if } Y < y^{0} \\ Y & \text{ if } Y > y^{0} \end{array} \right . \end{equation} where $\alpha$ and $\beta$ are known constants such that the mapping is continuous, but not differentiable at $Y = y^{0}$. How do I find the density function of $Y^{*}$?
My concern is how the non-differentiable point $Y=y^0$ affects the resulting density of $Y^∗$. My solution for the density of $Y^∗$ is the following:
$f_{Y^{*}}(y^{*}) = f_{Y}(y^{*}) I\{ Y^{*} > y^{0} \} + \frac{1}{\alpha}f_{Y}\left ( \frac{y^{*} - \beta}{\alpha} \right) I\{ Y^{*} < \alpha y^{0} + \beta \}. $
[self-study]
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