# Probability density function of $\sigma X + \mu$.

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?

-
For one thing, you expect that $\int_{\mathbb R} f_Z(x) \, dx = 1$, but if you use the formula for $f_Z(x)$ that you have, you don't get 1. – echoone Dec 4 '12 at 2:57
@echoone And why not? – Sasha Dec 4 '12 at 3:08
@echoone $\int_{\mathbb{R}} f_Z(x)dx = \int_{\mathbb{R}} f_X(x)dx = 1$. Don't forget the Jacobian when you change variables. – nullUser Dec 4 '12 at 3:09
Oops. Yeah, you guys are right. What I am wondering now is how you are dealing with the $\sigma < 0$ case. – echoone Dec 4 '12 at 3:37

1
– Learner Dec 4 '12 at 3:44
show 1 more comment
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9176804423332214, "perplexity_flag": "head"}