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Let $Z \sim N(\mu,1)$, $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent random variables.

By definition, the absolute value of a Normal random variable is said to have a folded Normal distribution.

I'm trying to find: $\; P(|Z| >|Y|) \;$ and $ \; P(|Z| >|X|)$.

Also, I'm trying to find, for the function: $\max[|Z|, |Y|, |X| ]$, what is the probability that |Z| is the max value of the three?

Reference also: Sum of Independent Folded-Normal distributions

Diana
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  • Welcome, please [edit] the post to add a link to the question you're referring to and use MathJax to type in mathematics. –  Apr 12 '17 at 15:58
  • (a) You need to define what $X$, $Y$ and $Z$ are - not sufficient to say half-folded --- you need to specify what parameters are included, if any. And this should be stated at the start - not at the end. (b) Probability that $Z$ has max value or that $|Z|$ has max value? (c) If some of your variables are folded Normals, then you don't need to take the absolute value, since the folding does that already. So you need to think about your notation. – wolfies Apr 12 '17 at 16:37
  • I updated my question based on your comments. Thanks – Diana Apr 13 '17 at 08:46
  • They don't have the same distribution, only y and x – Diana Apr 13 '17 at 16:32
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    Why do you need to find $; P(|Z| >|Y|) ;$ and $; P(|Z| >|X|) ;$ ... if $X$ and $Y$ have the same distribution? – wolfies Apr 13 '17 at 16:59
  • O.k. they are equal. then I can find the probability of |Z| is more than |X| and |Y|, i.e. P(|Z|>|Y|) * P(|Z|>|X|) – Diana Apr 14 '17 at 04:53

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