Suppose we have $0\le\,X\le\,\infty$, and we have $Y = a+b\, X\,\quad$ where $a\in\mathbb{R}$, $b\in\mathbb{R}$, $X \sim \chi^2_\nu(\beta)\quad$ with a PDF $f_X(x)$. Now, I found that the PDF of Y is given by
$f_Y(y) = \frac{1}{b}f_X\left(\frac{y-a}{b}\right)$
I have two questions please....
- is the formula right for $f_Y(y)$.
- if I want to obtain the expectation of another function of $Y$, given by $g(y)$, is it right to use
$\int^\infty_0 g(y)f_Y(y)\mathrm{d}y$
or $\int^\infty_a g(y)f_Y(y)\mathrm{d}y$.
Thanks.