Show how Gaussian with mean $\mu$ and std dev $\sigma$ is constructed.

Question

I am a computer science student, who has recently taken an interest in data science. I've been learning about Gaussian distributions, and I've read from the documentation of the numpy python package here that sampling from a Gaussian distribution parameterized by mean $\mu$ and standard deviation $\sigma$ is effectively the same as sampling from a "standard normal" distribution (mean 0, standard deviation 1), then multiplying the result by $\sigma$ and adding $\mu$. Can someone please provide a proof or explanation of why this is true? Please clarify any steps thaty may not be immediately obvious to a non-math student. Thanks

Answer 1

Special about normal distribution is: if $Y$ has normal distribution and $Z:=aY+b$ (where $a,b$ are constants and $a\neq0$) then also $Z$ has normal distribution.

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If we look at non-degenerate cases (as we mostly do) then the standard deviation $\sigma$ is positive. If $X$ denotes a random variable then based on it we can define a new one: $$U:=\frac{X-\mu}{\sigma}\tag1$$ The relation can also be expressed as: $$X=\sigma U+\mu\tag2$$

This works for all non-degenerate random variables that have a well defined standard deviation.

In the special case where $X$ has normal distribution we can apply what I wrote above the line to find that (1) implies that $U$ standard normal distribution.

Looking at (2) we see that any normal distributed $X$ can be written as a linear form of a standard normal distributed $U$.