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I need a head check on this one. Suppose $\sigma,\mu \in \mathbb{R}$ and $\sigma \neq 0$. Let $X$ be a random variable with density $f_X(x)$. I think that the random variable $Z:= \sigma X + \mu$ has density $ f_Z(x) = \frac{1}{|\sigma|}f_X\left(\frac{x-\mu}{\sigma} \right). $

Splitting the cases $\sigma > 0$ and $\sigma<0$ my proof comes down to a simple change of variables. However, I can't find any mention of a formula like this on wikipedia or google. Is it so simple that no one thought to mention it, or am I misunderstanding?

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    Another version of the formula is at http://en.wikipedia.org/wiki/Location-scale_family#Converting_a_single_distribution_to_a_location-scale_family2012-12-04

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