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I want to prove that if X is a random variable with the uniform distribution over [L, R] and Y = cX + d with c > 0, then the uniform distribution of Y is over the interval [cL + d, cR + d].

I'm using the following theorem:

$ f_Y(y) = \frac {f_X(h^{-1}(y))}{|(h'(h^{-1}(y))|} $

and I get:

$ f_Y(y) = \frac {f_X(\frac{y-d}{c})}{c} $, (*)

but I think I should be getting

$ f_Y(y) = \frac {f_X(\frac{y-d}{c})}{(R-L)c} $ (**)

to make my proof work. How do I reach (**) instead of (*)?

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(*) is correct, but you need to go one more step. Since, $f_X(x) = \dfrac 1{R-L}\;\mathbf 1_{x\in[L;R]}$

$$\begin{align}f_Y(y) & = \frac {f_X(\frac{y-d}{c})}{c} \\[2ex] & = \frac{1}{c(R-L)}\;\mathbf 1_{y\in[cL+d;cR+d]}\end{align}$$

That is all.

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