Ben Shaffer

2022-03-16

Calculating probabilities for complex random variables

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 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:

Brock Floyd

Beginner2022-03-17Added 6 answers

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

So$P\left(Y>0\right)$ is equivalent to $P\left(X>\frac{-2-\mu}{\sigma}\right)=1-\mathrm{\Phi}\left(\frac{-2-\mu}{\sigma}\right)$

So

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