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I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: $Y=10(2+\mu+\sigma X)$ where $X$ has standard normal distribution. How would I, for example, formulate/define $P(Y>0)$ for some $\mu$ and $\sigma$?

I am thinking, since $X\sim N(0,1)$ I could use the p.d.f of the standard normal distribution to find the probabilities. But I fail to understand on how I would plugin the values into $\Phi$, for example what happens with the constant $20$? Any elaboration on this would be appreciated.

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  • $\begingroup$ When you say "complex" do you mean complicated or something involving $\sqrt{-1}$? $\endgroup$
    – Henry
    Apr 16, 2015 at 9:54
  • $\begingroup$ I mean complicated. Sorry about that $\endgroup$
    – optimus257
    Apr 16, 2015 at 9:56

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Assuming $\sigma \gt 0$, from $Y=10(2+\mu+\sigma X)$ you could say $X = \dfrac{Y-20-10 \mu}{10\sigma}$.

So $P(Y \gt 0)$ is equivalent to $P\left(X \gt \dfrac{-2- \mu}{\sigma} \right) = 1 - \Phi\left(\dfrac{-2- \mu}{\sigma} \right)$.

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  • $\begingroup$ Just a small correction. You divide by $10\sigma$ not just $\sigma$ $\endgroup$
    – optimus257
    Apr 19, 2015 at 10:53
  • $\begingroup$ @optimus257: indeed - I have made the correction $\endgroup$
    – Henry
    Apr 19, 2015 at 18:55

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