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When reading the explanation for the "The reparametrization trick" on the Stanford's cs228 notes, I saw a claim that

It is easy to check that the two ways of expressing the random variable z lead to the same distribution.

See here:

enter image description here

(Just search or scroll down on that page to the section on reparametrization on that page to see the claim)

I'm not entirely sure how I'd go about checking that the two ways lead to the same distribution - how would I go about doing that?

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blueberryfields
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    Are you asking about specifically comparing $Z_1 \sim N(\mu, \sigma^2)$ and $Z_2=\mu + \epsilon \sigma$ where $\epsilon \sim N(0,1)$ or something more general? – Henry Aug 01 '23 at 22:34
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    @blueberryfields The easy way to understand that part would be purely algebraic, e.g. one approach would be by equality of two expressions for the cdf or pdf – Glen_b Aug 01 '23 at 23:02
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    If it is "the same random variable", the distribution is obviously the "same". – Xi'an Aug 01 '23 at 23:17
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    It's simple algebra, not even involving any statistics or probability concepts: because $\epsilon=(Z_2-\mu)/\sigma,$ it's immediate that $Z_2 = \mu+\epsilon\sigma,$ whence the two sides have the same distribution. – whuber Aug 01 '23 at 23:20
  • @whuber where do you see that on the reference page? Where did you get that equation for ? Are we looking at the same source, or am I missing something obvious? – blueberryfields Aug 03 '23 at 22:56
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    I am applying basic algebra (such applications are rarely given explicitly once the readers get beyond beginning algebra textbooks, and I think that's fair to assume of any class at Stanford) and a simple characterization of Normal distributions, nothing more. One effective characterization is the one I explain at https://stats.stackexchange.com/a/564628/919 ("all Normal distributions have the same shape"), but many other characterizations will work as well. – whuber Aug 03 '23 at 22:58
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    Aha. Right. My stats gears are rusty. Does the source have a typo then? Should it read 1∼(,^2)? – blueberryfields Aug 03 '23 at 23:15
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    Conventions differ. The only way to tell whether that's a typo is to review the entire text to see whether it explicitly defines that notation. When I scan below your quotation I see repeated instances where the second parameter is explicitly squared, so I believe you are correct. – whuber Aug 04 '23 at 12:52

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